# Knight's tour

(Redirected from Knight's Tour)
Knight's tour
You are encouraged to solve this task according to the task description, using any language you may know.

Problem: you have a standard 8x8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the tour be "closed"; that is, the knight need not end within a single move of its start position.

Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in algebraic notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered according to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.

Input: starting square

Output: move sequence

## 11l

Translation of: Python
```V _kmoves = [(2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)]

F chess2index(=chess, boardsize)
‘Convert Algebraic chess notation to internal index format’
chess = chess.lowercase()
V x = chess[0].code - ‘a’.code
V y = boardsize - Int(chess[1..])
R (x, y)

F boardstring(board, boardsize)
V r = 0 .< boardsize
V lines = ‘’
L(y) r
lines ‘’= "\n"r.map(x -> (I @board[(x, @y)] {‘#2’.format(@board[(x, @y)])} E ‘  ’)).join(‘,’)
R lines

F knightmoves(board, P, boardsize)
V (Px, Py) = P
V kmoves = Set(:_kmoves.map((x, y) -> (@Px + x, @Py + y)))
kmoves = Set(Array(kmoves).filter((x, y) -> x C 0 .< @boardsize & y C 0 .< @boardsize & !@board[(x, y)]))
R kmoves

F accessibility(board, P, boardsize)
[(Int, (Int, Int))] access
V brd = copy(board)
L(pos) knightmoves(board, P, boardsize' boardsize)
brd[pos] = -1
access.append((knightmoves(brd, pos, boardsize' boardsize).len, pos))
brd[pos] = 0
R access

F knights_tour(start, boardsize, _debug = 0B)
[(Int, Int) = Int] board
L(x) 0 .< boardsize
L(y) 0 .< boardsize
board[(x, y)] = 0
V move = 1
V P = chess2index(start, boardsize)
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
L move <= board.len
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move++
I _debug
print(boardstring(board, boardsize' boardsize))
input("\n#2 next: ".format(move))
R board

L(boardsize, start) [(5, ‘c3’), (8, ‘h8’), (10, ‘e6’)]
print(‘boardsize: ’boardsize)
print(‘Start position: ’start)
V board = knights_tour(start, boardsize)
print(boardstring(board, boardsize' boardsize))
print()```
Output:
```boardsize: 5
Start position: c3

19,12,17, 6,21
2, 7,20,11,16
13,18, 1,22, 5
8, 3,24,15,10
25,14, 9, 4,23

boardsize: 8
Start position: h8

38,41,18, 3,22,27,16, 1
19, 4,39,42,17, 2,23,26
40,37,54,21,52,25,28,15
5,20,43,56,59,30,51,24
36,55,58,53,44,63,14,29
9, 6,45,62,57,60,31,50
46,35, 8,11,48,33,64,13
7,10,47,34,61,12,49,32

boardsize: 10
Start position: e6

29, 4,57,24,73, 6,95,10,75, 8
58,23,28, 5,94,25,74, 7,100,11
3,30,65,56,27,72,99,96, 9,76
22,59, 2,63,68,93,26,81,12,97
31,64,55,66, 1,82,71,98,77,80
54,21,60,69,62,67,92,79,88,13
49,32,53,46,83,70,87,42,91,78
20,35,48,61,52,45,84,89,14,41
33,50,37,18,47,86,39,16,43,90
36,19,34,51,38,17,44,85,40,15

```

## 360 Assembly

Translation of: BBC PASIC
```*        Knight's tour             20/03/2017
KNIGHT   CSECT
USING  KNIGHT,R13         base registers
B      72(R15)            skip savearea
DC     17F'0'             savearea
STM    R14,R12,12(R13)    save previous context
ST     R13,4(R15)         link backward
ST     R15,8(R13)         link forward
LR     R13,R15            set addressability
MVC    PG(20),=CL20'Knight''s tour ..x..'
L      R1,NN              n
XDECO  R1,XDEC            edit
MVC    PG+14(2),XDEC+10   n
MVC    PG+17(2),XDEC+10   n
XPRNT  PG,L'PG            print buffer
LA     R0,1               1
ST     R0,X               x=1
ST     R0,Y               y=1
SR     R0,R0              0
ST     R0,TOTAL           total=0
LOOP     EQU    *                  do loop
L      R1,X                 x
BCTR   R1,0                 -1
MH     R1,NNH               *n
L      R0,Y                 y
BCTR   R0,0                 -1
AR     R1,R0                (x-1)*n+y-1
SLA    R1,1                 ((x-1)*n+y-1)*2
LA     R0,1                 1
STH    R0,BOARD(R1)         board(x,y)=1
L      R2,TOTAL             total
LA     R2,1(R2)             total+1
STH    R2,DISP(R1)          disp(x,y)=total+1
ST     R2,TOTAL             total=total+1
L      R1,X                 x
L      R2,Y                 y
BAL    R14,CHOOSEMV         call choosemv(x,y)
C      R0,=F'0'           until(choosemv(x,y)=0)
BNE    LOOP               loop
LA     R2,KN*KN           n*n
IF C,R2,NE,TOTAL THEN       if total<>n*n then
XPRNT  =C'error!!',7        print error
ENDIF    ,                  endif
LA     R6,1               i=1
DO WHILE=(C,R6,LE,NN)       do i=1 to n
MVC    PG,=CL128' '         init buffer
LA     R10,PG               pgi=0
LA     R7,1                 j=1
DO WHILE=(C,R7,LE,NN)         do j=1 to n
LR     R1,R6                  i
BCTR   R1,0                   -1
MH     R1,NNH                 *n
LR     R0,R7                  j
BCTR   R0,0                   -1
AR     R1,R0                  (i-1)*n+j-1
SLA    R1,1                   ((i-1)*n+j-1)*2
LH     R2,DISP(R1)            disp(i,j)
XDECO  R2,XDEC                edit
MVC    0(4,R10),XDEC+8        output
LA     R10,4(R10)             pgi+=4
LA     R7,1(R7)               j++
ENDDO    ,                    enddo j
XPRNT  PG,L'PG              print buffer
LA     R6,1(R6)             i++
ENDDO    ,                  enddo i
L      R13,4(0,R13)       restore previous savearea pointer
LM     R14,R12,12(R13)    restore previous context
XR     R15,R15            return_code=0
BR     R14                exit
*------- ----   ----------------------------------------
CHOOSEMV EQU    *                  choosemv(xc,yc)
ST     R14,SAVEACMV       save return point
ST     R1,XC              store xc
ST     R2,YC              store yc
MVC    MM,=F'9'           m=9
L      R1,XC              xc
LA     R1,1(R1)
L      R2,YC              yc
LA     R2,2(R2)
BAL    R14,TRYMV          call trymv(xc+1,yc+2)
L      R1,XC              xc
LA     R1,1(R1)
L      R2,YC              yc
SH     R2,=H'2'
BAL    R14,TRYMV          call trymv(xc+1,yc-2)
L      R1,XC              xc
BCTR   R1,0
L      R2,YC              yc
LA     R2,2(R2)
BAL    R14,TRYMV          call trymv(xc-1,yc+2)
L      R1,XC              xc
BCTR   R1,0
L      R2,YC              yc
SH     R2,=H'2'
BAL    R14,TRYMV          call trymv(xc-1,yc-2)
L      R1,XC              xc
LA     R1,2(R1)
L      R2,YC              yc
LA     R2,1(R2)
BAL    R14,TRYMV          call trymv(xc+2,yc+1)
L      R1,XC              xc
LA     R1,2(R1)
L      R2,YC              yc
BCTR   R2,0
BAL    R14,TRYMV          call trymv(xc+2,yc-1)
L      R1,XC              xc
SH     R1,=H'2'
L      R2,YC              yc
LA     R2,1(R2)
BAL    R14,TRYMV          call trymv(xc-2,yc+1)
L      R1,XC              xc
SH     R1,=H'2'
L      R2,YC              yc
BCTR   R2,0
BAL    R14,TRYMV          call trymv(xc-2,yc-1)
L      R4,MM              m
IF C,R4,EQ,=F'9' THEN       if m=9 then
LA     R0,0                 return(0)
ELSE     ,                  else
MVC    X,NEWX               x=newx
MVC    Y,NEWY               y=newy
LA     R0,1                 return(1)
ENDIF    ,                  endif
L      R14,SAVEACMV       restore return point
BR     R14                return
SAVEACMV DS     A                  return point
*------- ----   ----------------------------------------
TRYMV    EQU    *                  trymv(xt,yt)
ST     R14,SAVEATMV       save return point
ST     R1,XT              store xt
ST     R2,YT              store yt
SR     R10,R10            n=0
BAL    R14,VALIDMV
IF LTR,R0,Z,R0 THEN         if validmv(xt,yt)=0 then
LA     R0,0                 return(0)
B      RETURTMV
ENDIF    ,                  endif
L      R1,XT
LA     R1,1(R1)           xt+1
L      R2,YT
LA     R2,2(R2)           yt+2
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt+1,yt+2)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
LA     R1,1(R1)           xt+1
L      R2,YT
SH     R2,=H'2'           yt-2
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt+1,yt-2)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
BCTR   R1,0               xt-1
L      R2,YT
LA     R2,2(R2)           yt+2
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt-1,yt+2)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
BCTR   R1,0               xt-1
L      R2,YT
SH     R2,=H'2'           yt-2
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt-1,yt-2)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
LA     R1,2(R1)           xt+2
L      R2,YT
LA     R2,1(R2)           yt+1
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt+2,yt+1)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
LA     R1,2(R1)           xt+2
L      R2,YT
BCTR   R2,0               yt-1
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt+2,yt-1)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
SH     R1,=H'2'           xt-2
L      R2,YT
LA     R2,1(R2)           yt+1
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt-2,yt+1)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
L      R1,XT
SH     R1,=H'2'           xt-2
L      R2,YT
BCTR   R2,0               yt-1
BAL    R14,VALIDMV
IF C,R0,EQ,=F'1' THEN       if validmv(xt-2,yt-1)=1 then
LA     R10,1(R10)           n=n+1;
ENDIF    ,                  endif
IF C,R10,LT,MM THEN         if n<m then
ST     R10,MM               m=n
MVC    NEWX,XT              newx=xt
MVC    NEWY,YT              newy=yt
ENDIF    ,                  endif
RETURTMV L      R14,SAVEATMV       restore return point
BR     R14                return
SAVEATMV DS     A                  return point
*------- ----   ----------------------------------------
VALIDMV  EQU    *                  validmv(xv,yv)
C      R1,=F'1'           if xv<1  then
BL     RET0
C      R1,NN              if xv>nn then
BH     RET0
C      R2,=F'1'           if yv<1  then
BL     RET0
C      R2,NN              if yv>nn then
BNH    OK
RET0     SR     R0,R0              return(0)
B      RETURVMV
OK       LR     R3,R1              xv
BCTR   R3,0
MH     R3,NNH             *n
LR     R0,R2              yv
BCTR   R0,0
AR     R3,R0
SLA    R3,1
LH     R4,BOARD(R3)       board(xv,yv)
IF LTR,R4,Z,R4 THEN         if board(xv,yv)=0 then
LA     R0,1                 return(1)
ELSE     ,                  else
SR     R0,R0                return(0)
ENDIF    ,                  endif
RETURVMV BR     R14                return
*        ----   ----------------------------------------
KN       EQU    8                  n  compile-time
NN       DC     A(KN)              n  fullword
NNH      DC     AL2(KN)            n  halfword
BOARD    DC     (KN*KN)H'0'        dim board(n,n) init 0
DISP     DC     (KN*KN)H'0'        dim  disp(n,n) init 0
X        DS     F
Y        DS     F
TOTAL    DS     F
XC       DS     F
YC       DS     F
MM       DS     F
NEWX     DS     F
NEWY     DS     F
XT       DS     F
YT       DS     F
XDEC     DS     CL12
PG       DC     CL128' '           buffer
YREGS
END    KNIGHT```
Output:
```Knight's tour  8x 8
1   4  57  20  47   6  49  22
34  19   2   5  58  21  46   7
3  56  35  60  37  48  23  50
18  33  38  55  52  59   8  45
39  14  53  36  61  44  51  24
32  17  40  43  54  27  62   9
13  42  15  30  11  64  25  28
16  31  12  41  26  29  10  63
```

First, we specify a naive implementation the package Knights_Tour with naive backtracking. It is a bit more general than required for this task, by providing a mechanism not to visit certain coordinates. This mechanism is actually useful for the task Solve a Holy Knight's tour#Ada, which also uses the package Knights_Tour.

```generic
Size: Integer;
package Knights_Tour is

subtype Index is Integer range 1 .. Size;
type Tour is array  (Index, Index) of Natural;
Empty: Tour := (others => (others => 0));

function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty) return Tour;
-- finds tour via backtracking
-- either no tour has been found, i.e., Get_Tour returns Scene
-- or the Result(X,Y)=K if and only if I,J is visited at the K-th move
-- for all X, Y, Scene(X,Y) must be either 0 or Natural'Last,
--   where Scene(X,Y)=Natural'Last means "don't visit coordiates (X,Y)!"

function Count_Moves(Board: Tour) return Natural;
-- counts the number of possible moves, i.e., the number of 0's on the board

procedure Tour_IO(The_Tour: Tour; Width: Natural := 4);
-- writes The_Tour to the output using Ada.Text_IO;

end Knights_Tour;
```

Here is the implementation:

```with Ada.Text_IO, Ada.Integer_Text_IO;

package body Knights_Tour is

type Pair is array(1..2) of Integer;
type Pair_Array is array (Positive range <>) of Pair;

Pairs: constant Pair_Array (1..8)
:= ((-2,1),(-1,2),(1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1));
-- places for the night to go (relative to the current position)

function Count_Moves(Board: Tour) return Natural is
N: Natural := 0;
begin
for I in Index loop
for J in Index loop
if Board(I,J) < Natural'Last then
N := N + 1;
end if;
end loop;
end loop;
return N;
end Count_Moves;

function Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
Done: Boolean;
Move_Count: Natural := Count_Moves(Scene);
Visited: Tour;

-- Visited(I, J) = 0: not yet visited
-- Visited(I, J) = K: visited at the k-th move
-- Visited(I, J) = Integer'Last: never visit

procedure Visit(X, Y: Index; Move_Number: Positive; Found: out Boolean) is
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move_Number;
if Move_Number = Move_Count then
Found := True;
else
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Visit(XX, YY, Move_Number+1, Found); -- recursion
if Found then
return; -- no need to search further
end if;
end if;
end loop;
Visited(X, Y) := 0; -- undo previous mark
end if;
end Visit;

begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Get_Tour;

procedure Tour_IO(The_Tour: Tour; Width: Natural := 4) is
begin
for I in Index loop
for J in Index loop
if The_Tour(I, J) < Integer'Last then
else
for W in 1 .. Width-1 loop
end loop;
Ada.Text_IO.Put("-"); -- deliberately not visited
end if;
end loop;
end loop;
end Tour_IO;

end Knights_Tour;
```

Here is the main program:

```with Knights_Tour, Ada.Command_Line;

procedure Test_Knight is

Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));

package KT is new Knights_Tour(Size => Size);

begin
KT.Tour_IO(KT.Get_Tour(1, 1));
end Test_Knight;
```

For small sizes, this already works well (< 1 sec for size 8). Sample output:

```>./test_knight 8
1  38  55  34   3  36  19  22
54  47   2  37  20  23   4  17
39  56  33  46  35  18  21  10
48  53  40  57  24  11  16   5
59  32  45  52  41  26   9  12
44  49  58  25  62  15   6  27
31  60  51  42  29   8  13  64
50  43  30  61  14  63  28   7```

For larger sizes we'll use Warnsdorff's heuristic (without any thoughtful tie breaking). We enhance the specification adding a function Warnsdorff_Get_Tour. This enhancement of the package Knights_Tour will also be used for the task Solve a Holy Knight's tour#Ada. The specification of Warnsdorff_Get_Tour is the following.

```
function Warnsdorff_Get_Tour(Start_X, Start_Y: Index; Scene: Tour := Empty)
-- uses Warnsdorff heurisitic to find a tour faster
-- same interface as Get_Tour
```

Its implementation is as follows.

```   function Warnsdorff_Get_Tour(Start_X, Start_Y: Index;  Scene: Tour := Empty)
Done: Boolean;
Visited: Tour; -- see comments from Get_Tour above
Move_Count: Natural := Count_Moves(Scene);

function Neighbors(X, Y: Index) return Natural is
Result: Natural := 0;
begin
for P in Pairs'Range loop
if X+Pairs(P)(1) in Index and then Y+Pairs(P)(2) in Index and then
Visited(X+Pairs(P)(1),  Y+Pairs(P)(2)) = 0 then
Result := Result + 1;
end if;
end loop;
return Result;
end Neighbors;

procedure Sort(Options: in out Pair_Array) is
N_Bors: array(Options'Range) of Natural;
K: Positive range Options'Range;
N: Natural;
P: Pair;
begin
for Opt in Options'Range loop
N_Bors(Opt) := Neighbors(Options(Opt)(1), Options(Opt)(2));
end loop;
for Opt in Options'Range loop
K := Opt;
for Alternative in Opt+1 .. Options'Last loop
if N_Bors(Alternative) < N_Bors(Opt) then
K := Alternative;
end if;
end loop;
N           := N_Bors(Opt);
N_Bors(Opt) := N_Bors(K);
N_Bors(K)   := N;
P            := Options(Opt);
Options(Opt) := Options(K);
Options(K)   := P;
end loop;
end Sort;

procedure Visit(X, Y: Index; Move: Positive; Found: out Boolean) is
Next_Count: Natural range 0 .. 8 := 0;
Next_Steps: Pair_Array(1 .. 8);
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move;
if Move = Move_Count then
Found := True;
else
-- consider all possible places to go
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Next_Count := Next_Count+1;
Next_Steps(Next_Count) := (XX, YY);
end if;
end loop;

Sort(Next_Steps(1 .. Next_Count));

for N in 1 .. Next_Count loop
Visit(Next_Steps(N)(1), Next_Steps(N)(2), Move+1, Found);
if Found then
return; -- no need to search further
end if;
end loop;

-- if we didn't return above, we have to undo our move
Visited(X, Y) := 0;
end if;
end Visit;

begin
Visited := Scene;
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := Scene;
end if;
return Visited;
end Warnsdorff_Get_Tour;
```

The modification for the main program is trivial:

```with Knights_Tour, Ada.Command_Line;

procedure Test_Fast is

Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));

package KT is new Knights_Tour(Size => Size);

begin
KT.Tour_IO(KT.Warnsdorff_Get_Tour(1, 1));
end Test_Fast;
```

This works still well for somewhat larger sizes:

```>./test_fast 24
1 108  45  52   3 112  57  60   5  62 131 144   7  64 147 170   9  66 187 192  11  68  71 190
46  51   2 111  56  53   4 113 130  59   6  63 146 169   8  65 186 215  10  67 188 191  12  69
107  44 109  54 123 114 129  58  61 132 145 168 143 148 185 214 171 198 225 216 193  70 189  72
50  47 122 115 110  55 140 133 128 167 142 149 184 213 172 199 226 255 246 197 224 217 194  13
43 106  49 124 139 134 127 166 141 150 183 212 173 200 227 254 247 242 223 256 245 196  73 218
48 121 116 135 126 165 138 151 182 211 174 201 228 253 248 241 290 263 304 243 222 257  14 195
105  42 125 164 137 152 181 210 175 202 229 252 249 240 289 264 329 308 291 262 303 244 219  74
120 117 136 153 180 163 176 203 230 267 250 239 288 265 328 309 334 345 330 305 292 221 258  15
41 104 119 160 177 204 231 268 209 238 287 266 251 310 335 344 357 332 307 346 261 302  75 220
118 159 154 205 162 179 208 237 286 269 324 311 336 327 438 333 418 347 356 331 306 293  16 259
103  40 161 178 207 232 285 270 323 312 337 326 483 416 343 422 437 358 419 298 349 260 301  76
158 155 206 233 284 271 236 313 338 325 482 415 342 439 484 417 420 423 348 355 360 299 294  17
39 102 157 272 235 314 339 322 481 414 341 492 497 514 421 440 485 436 359 424 297 350  77 300
156 273 234 315 276 283 478 413 340 493 480 513 530 491 498 515 452 441 454 435 354 361  18 295
101  38 275 282 397 412 321 494 479 512 557 496 543 534 529 490 499 486 451 442 425 296 351  78
274 279 316 277 320 477 410 511 570 495 554 535 556 531 542 533 516 453 444 455 434 353 362  19
37 100 281 398 411 396 575 476 567 558 561 544 553 536 521 528 489 500 487 450 443 426  79 352
280 317 278 319 402 409 510 569 560 571 566 555 550 541 532 537 522 517 460 445 456 433  20 363
99  36 389 378 399 576 395 574 475 568 559 562 545 552 525 520 527 488 501 462 449 364 427  80
94 379 318 401 388 403 408 509 572 565 474 551 540 549 538 523 518 461 446 459 432 457 366  21
35  98  93 390 377 400 573 394 375 508 563 546 373 524 519 526 371 502 463 466 365 448  81 428
380  95 382 385 404 387 376 407 564 473 374 507 548 539 372 503 464 467 370 447 458 431  22 367
383  34  97  92 391  32 405  90 393  30 547  88 471  28 505  86 469  26 465  84 369  24 429  82
96 381 384  33 386  91 392  31 406  89 472  29 506  87 470  27 504  85 468  25 430  83 368  23```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32
```# Non-recursive Knight's Tour with Warnsdorff's algorithm                #
# If there are multiple choices, backtrack if the first choice doesn't   #
# find a solution                                                        #

# the size of the board                                                  #
INT board size = 8;

# directions for moves #
INT nne = 1, nee = 2, see = 3, sse = 4, ssw = 5, sww = 6, nww = 7, nnw = 8;

INT lowest move  = nne;
INT highest move = nnw;

# the vertical position changes of the moves                             #
#                  nne, nee, see, sse, ssw, sww, nww, nnw                #
[]INT offset v = (  -2,  -1,   1,   2,   2,   1,  -1,  -2 );
# the horizontal position changes of the moves                           #
#                  nne, nee, see, sse, ssw, sww, nww, nnw                #
[]INT offset h = (   1,   2,   2,   1,  -1,  -2,  -2,  -1 );

MODE SQUARE = STRUCT( INT move      # the number of the move that caused #
# the knight to reach this square    #
, INT direction # the direction of the move that     #
# brought the knight here - one of   #
# nne, nee, see, sse, ssw, sww, nww  #
# or nnw - used for backtracking     #
# zero for the first move            #
);

# the board #
[ board size, board size ]SQUARE board;

# initialises the board so there are no used squares #
PROC initialise board = VOID:
FOR row FROM 1 LWB board TO 1 UPB board
DO
FOR col FROM 2 LWB board TO 2 UPB board
DO
board[ row, col ] := ( 0, 0 )
OD
OD; # initialise board #

INT iterations := 0;
INT backtracks := 0;

# prints the board #
PROC print tour = VOID:
BEGIN

print( ( "       a   b   c   d   e   f   g   h", newline ) );
print( ( "   +--------------------------------", newline ) );

FOR row FROM 1 UPB board BY -1 TO 1 LWB board
DO
print( ( whole( row, -3 ) ) );
print( ( "|" ) );

FOR col FROM 2 LWB board TO 2 UPB board
DO
print( ( " " ) );
print( ( whole( move OF board[ row, col ], -3 ) ) )
OD;
print( ( newline ) )
OD

END; # print tour #

# determines whether a move to the specified row and column is possible #
PROC can move to = ( INT row, INT col )BOOL:
IF row > 1 UPB board
OR row < 1 LWB board
OR col > 2 UPB board
OR col < 2 LWB board
THEN
# the position is not on the board                              #
FALSE
ELSE
# the move is legal, check the square is unoccupied             #
move OF board[ row, col ] = 0
FI;

# used to hold counts of the number of moves that could be made in each #
# direction from the current square                                     #
[ lowest move : highest move ]INT possible move count;

# sets the elements of possible move count to the number of moves that  #
# could be made in each direction from the specified row and col        #
PROC count moves in each direction from = ( INT row, INT col )VOID:
FOR move direction FROM lowest move TO highest move
DO

INT new row = row + offset v[ move direction ];
INT new col = col + offset h[ move direction ];

IF NOT can move to( new row, new col )
THEN
# can't move to this square #
possible move count[ move direction ] := -1
ELSE
# a move in this direction is possible #
# - count the number of moves that could be made from it #

possible move count[ move direction ] := 0;

FOR subsequent move FROM lowest move TO highest move
DO
IF can move to( new row + offset v[ subsequent move ]
, new col + offset h[ subsequent move ]
)
THEN
# have a possible subsequent move #
possible move count[ move direction ] +:= 1
FI
OD
FI

OD;

# update the board to the first knight's tour found starting from       #
# "start row" and "start col".                                          #
# return TRUE if one was found, FALSE otherwise                         #
PROC find tour = ( INT start row, INT start col )BOOL:
BEGIN

initialise board;

BOOL result := TRUE;

INT  move number  := 1;
INT  row          := start row;
INT  col          := start col;

# the tour will be complete when we have made as many moves            #
# as there squares on the board                                        #
INT  final move    = ( ( ( 1 UPB board ) + 1 ) - 1 LWB board )
* ( ( ( 2 UPB board ) + 1 ) - 2 LWB board )
;

# the first move is to place the knight on the starting square         #
board[ row, col ]  := ( move number, lowest move - 1 );
# start off with an unknown direction for the best move                #
INT best direction := lowest move - 1;

# attempt to find a sequence of moves that will reach each square once #
WHILE
move number < final move AND result
DO

iterations +:= 1;

# count the number of moves possible from each possible move       #
# from this square                                                 #
count moves in each direction from( row, col );

# find the direction with the lowest number of subsequent moves    #

IF best direction < lowest move
THEN
# must find the best direction to move in                      #

INT lowest move count := highest move + 1;

FOR move direction FROM lowest move TO highest move
DO
IF  possible move count[ move direction ] >= 0
AND possible move count[ move direction ] <  lowest move count
THEN
# have a move with fewer possible subsequent moves     #
best direction    := move direction;
lowest move count := possible move count[ move direction ]
FI
OD

ELSE
# following a backtrack - find an alternative with the same    #
# lowest number of possible moves - if there are any           #
# if there aren't, we will backtrack again                     #

INT lowest move count := possible move count[ best direction ];

WHILE
best direction +:= 1;
IF best direction > highest move
THEN
# no more possible moves with the lowest number of     #
# subsequent moves                                     #
FALSE
ELSE
# keep looking if the number of moves from this square #
# isn't the lowest                                     #
possible move count[ best direction ] /= lowest move count
FI
DO
SKIP
OD

FI;

IF best direction  <= highest move
AND best direction >= lowest move
THEN
# we found a best possible move #

INT new row = row + offset v[ best direction ];
INT new col = col + offset h[ best direction ];

row               := new row;
col               := new col;
move number      +:= 1;
board[ row, col ] := ( move number, best direction );

best direction    := lowest move - 1

ELSE
# no more moves from this position - backtrack #

IF move number = 1
THEN
# at the starting position - no solution #
result := FALSE

ELSE
# not at the starting position - undo the latest move #

backtracks  +:= 1;

move number -:= 1;

INT curr row := row;
INT curr col := col;

best direction := direction OF board[ curr row, curr col ];

row -:= offset v[ best direction ];
col -:= offset h[ best direction ];

# reset the square we just backtracked from #
board[ curr row, curr col ] := ( 0, 0 )

FI

FI

OD;

result
END; # find tour #

main:(

# get the starting position #

CHAR  row;
CHAR  col;

WHILE
print( ( "Enter starting row(1-8) and col(a-h): " ) );
read ( ( row, col, newline ) );
row < "1" OR row > "8" OR col < "a" OR col > "h"
DO
SKIP
OD;

# calculate the tour from that position, if possible #

IF find tour( ABS row - ABS "0", ( ABS col - ABS "a" ) + 1 )
THEN
# found a solution #
print tour
ELSE
# couldn't find a solution #
print( ( "Solution not found - iterations: ", iterations
, ", backtracks: ", backtracks
, newline
)
)
FI

)```
Output:
```Enter starting row(1-8) and col(a-h): 5d
a   b   c   d   e   f   g   h
+--------------------------------
8|  51  18  53  20  41  44   3   6
7|  54  21  50  45   2   5  40  43
6|  17  52  19  58  49  42   7   4
5|  22  55  64   1  46  57  48  39
4|  33  16  23  56  59  38  29   8
3|  24  13  34  63  30  47  60  37
2|  15  32  11  26  35  62   9  28
1|  12  25  14  31  10  27  36  61
```

## ATS

```(*
Find Knight’s Tours.

Using Warnsdorff’s heuristic, find multiple solutions.
Optionally accept only closed tours.

Compile with:
patscc -O3 -DATS_MEMALLOC_GCBDW -o knights_tour knights_tour.dats -lgc

Usage: ./knights_tour [START_POSITION [MAX_TOURS [closed]]]
Examples:
./knights_tour     (prints one tour starting from a1)
./knights_tour c5
./knights_tour c5 2000
./knights_tour c5 2000 closed
*)

#define ATS_DYNLOADFLAG 0       (* No initialization is needed. *)

#include "share/atspre_define.hats"

#define EMPTY_SQUARE ~1
macdef nil_move = @(~1, ~1)

fn
int_right_justified
{i : int}
{n : int | 0 <= n; n < 100}
(i : int i,
n : int n) :
string =
let
var buffer : @[char][100] = @[char][100] ('\0')
val _ = \$extfcall (int, "snprintf", buffer, 100, "%*i", n, i)
in
strnptr2string (string1_copy (\$UNSAFE.cast{string n} buffer))
end

typedef move_t (i : int,
j : int) =
@(int i, int j)
typedef move_t =
[i, j : int]
move_t (i, j)

fn
move_t_is_nil (move : move_t) :<>
bool =
let
val @(i, j) = move
val @(i_nil, j_nil) = nil_move
in
(i = i_nil && j = j_nil)
end

fn
move_t_fprint (f    : FILEref,
move : move_t) :
void =
let
val @(i, j) = move
val letter = char2i 'a' + j - 1
val digit = char2i '0' + i
in
fileref_putc (f, letter);
fileref_putc (f, digit);
end

vtypedef chessboard_vt (t       : t@ype,
n_ranks : int,
n_files : int,
p       : addr) =
@{
pf_board = @[t][n_ranks * n_files] @ p |
n_ranks = uint n_ranks,
n_files = uint n_files,
n_squares = uint (n_ranks * n_files),
p_board = ptr p
}
vtypedef chessboard_vt (t       : t@ype,
n_ranks : int,
n_files : int) =
chessboard_vt (t, n_ranks, n_files, p)
vtypedef chessboard_vt (t : t@ype) =
[n_ranks, n_files : int]
chessboard_vt (t, n_ranks, n_files)

fn {t : t@ype}
chessboard_vt_make
{n_ranks, n_files : pos}
(n_ranks : uint n_ranks,
n_files : uint n_files,
fill    : t) :
chessboard_vt (t, n_ranks, n_files) =
let
val size = u2sz (n_ranks * n_files)
val @(pf, pfgc | p) = array_ptr_alloc<t> (size)
val _ = array_initize_elt<t> (!p, size, fill)
prval _ = mfree_gc_v_elim pfgc (* Let the memory leak. *)
in
@{
pf_board = pf |
n_ranks = n_ranks,
n_files = n_files,
n_squares = n_ranks * n_files,
p_board = p
}
end

fn {t : t@ype}
chessboard_vt_get
{n_ranks, n_files : pos}
{i, j       : int}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i          : int i,
j          : int j) :
t =
let
val index = (i - 1) + (u2i (chessboard.n_ranks) * (j - 1))
val _ = assertloc (0 <= index)
val _ = assertloc (index < u2i (chessboard.n_squares))
in
array_get_at (!(chessboard.p_board), index)
end

fn {t : t@ype}
chessboard_vt_set
{n_ranks, n_files : pos}
{i, j       : int}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i          : int i,
j          : int j,
value      : t) :
void =
let
val index = (i - 1) + (u2i (chessboard.n_ranks) * (j - 1))
val _ = assertloc (0 <= index)
val _ = assertloc (index < u2i (chessboard.n_squares))
in
array_set_at (!(chessboard.p_board), index, value)
end

extern fn {t : t@ype}
find_nth_position\$equal (x : t,
y : t) :
bool

fn {t : t@ype}
find_nth_position
{n_ranks, n_files : pos}
(chessboard : !chessboard_vt (t, n_ranks, n_files),
n          : t) :
[i, j : int]
move_t (i, j) =
let
val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files

fun
outer_loop {i : pos | i <= n_ranks + 1} .<n_ranks + 1 - i>.
(chessboard : !chessboard_vt (t, n_ranks, n_files),
i : int i) :
[i, j : int]
move_t (i, j) =
let
fun
inner_loop {j : pos | j <= n_files + 1} .<n_files + 1 - j>.
(chessboard : !chessboard_vt (t, n_ranks, n_files),
j : int j) :
[j : int]
int j =
if u2i n_files < j then
j
else
let
val v = chessboard_vt_get<t> (chessboard, i, j)
in
if find_nth_position\$equal<t> (n, v) then
j
else
inner_loop (chessboard, succ j)
end
in
if u2i n_ranks < i then
nil_move
else
let
val j = inner_loop (chessboard, 1)
in
if j <= u2i n_files then
@(i, j)
else
outer_loop (chessboard, succ i)
end
end
in
outer_loop (chessboard, 1)
end

implement
find_nth_position\$equal<int> (x, y) =
x = y

fn
knights_tour_is_closed
{n_ranks, n_files : pos}
(chessboard : !chessboard_vt (int, n_ranks, n_files)) :
bool =
let
val n_squares = chessboard.n_squares
val @(i1, j1) = find_nth_position<int> (chessboard, 1)
val @(i2, j2) = find_nth_position<int> (chessboard, u2i n_squares)
val i_diff = abs (i1 - i2)
val j_diff = abs (j1 - j2)
in
(i_diff = 1 && j_diff = 2) || (i_diff = 2 && j_diff = 1)
end

fn
knights_tour_board_fprint
{n_ranks, n_files : pos}
(f          : FILEref,
chessboard : !chessboard_vt (int, n_ranks, n_files)) :
void =
{
val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files

fun
outer_loop {i : int | 0 <= i; i <= n_ranks} .<i>.
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i : int i) :
void =
if 0 < i then
{
val _ = fileref_puts (f, "    ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
fileref_puts (f, "+----")
end
val _ = fileref_puts (f, "+\n")
val _ = fileref_puts (f, int_right_justified (i, 2))
val _ = fileref_puts (f, " ")

fun
inner_loop {j : int | 1 <= j; j <= n_files + 1}
(chessboard : !chessboard_vt (int, n_ranks,
n_files),
j : int j) :
void =
if j <= u2i n_files then
{
val v = chessboard_vt_get<int> (chessboard, i, j)
val v = g1ofg0 v
val _ = fileref_puts (f, " | ")
val _ =
if v = EMPTY_SQUARE then
fileref_puts (f, "  ")
else
fileref_puts (f, int_right_justified (g1ofg0 v, 2))
val _ = inner_loop (chessboard, succ j)
}

val _ = inner_loop (chessboard, 1)
val _ = fileref_puts (f, " |\n")

val _ = outer_loop (chessboard, pred i)
}

val _ = outer_loop (chessboard, u2i n_ranks)
val _ = fileref_puts (f, "    ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
fileref_puts (f, "+----")
end
val _ = fileref_puts (f, "+\n")
val _ = fileref_puts (f, "   ")
val _ =
let
var j : [j : int] int j
in
for (j := 1; j <= u2i n_files; j := succ j)
let
val letter = char2i 'a' + j - 1
in
fileref_puts (f, "    ");
fileref_putc (f, letter)
end
end
}

fn
knights_tour_moves_fprint
{n_ranks, n_files : pos}
(f          : FILEref,
chessboard : !chessboard_vt (int, n_ranks, n_files)) :
void =
{
prval _ = mul_pos_pos_pos (mul_make {n_ranks, n_files} ())

val n_ranks = chessboard.n_ranks
val n_files = chessboard.n_files
val n_squares = chessboard.n_squares

val @(pf, pfgc | p_positions) =
array_ptr_alloc<move_t> (u2sz n_squares)
val _ = array_initize_elt<move_t> (!p_positions, u2sz n_squares,
nil_move)

macdef positions = !p_positions

fun
loop {k : int | 0 <= k; k <= n_ranks * n_files}
.<n_ranks * n_files - k>.
(positions  : &(@[move_t][n_ranks * n_files]),
chessboard : !chessboard_vt (int, n_ranks, n_files),
k          : int k) :
void =
if k < u2i n_squares then
{
val i = u2i ((i2u k) mod n_ranks) + 1
val j = u2i ((i2u k) / n_ranks) + 1
val v = chessboard_vt_get<int> (chessboard, i, j)
val v = g1ofg0 v
val _ = assertloc (1 <= v)
val _ = assertloc (v <= u2i n_squares)
val _ = positions[v - 1] := @(i, j)
val _ = loop (positions, chessboard, succ k)
}
val _ = loop (positions, chessboard, 0)

fun
loop {k : int | 0 <= k; k < n_ranks * n_files}
.<n_ranks * n_files - k>.
(positions : &(@[move_t][n_ranks * n_files]),
k         : int k) :
void =
if k < u2i (pred n_squares) then
{
val _ = move_t_fprint (f, positions[k])
val line_end = (((i2u (k + 1)) mod n_files) = 0U)
val _ =
fileref_puts (f, (if line_end then " ->\n" else " -> "))
val _ = loop (positions, succ k)
}
val _ = loop (positions, 0)
val _ = move_t_fprint (f, positions[pred n_squares])
val _ =
if knights_tour_is_closed (chessboard) then
fileref_puts (f, " -> cycle")

val _ = array_ptr_free (pf, pfgc | p_positions)
}

typedef knights_moves_t =
@(move_t, move_t, move_t, move_t,
move_t, move_t, move_t, move_t)

fn
possible_moves {n_ranks, n_files : pos}
{i, j       : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i          : int i,
j          : int j) :
knights_moves_t =
let
fn
try_move {istride, jstride : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
istride    : int istride,
jstride    : int jstride) :
move_t =
let
val i1 = i + istride
val j1 = j + jstride
in
if i1 < 1 then
nil_move
else if u2i (chessboard.n_ranks) < i1 then
nil_move
else if j1 < 1 then
nil_move
else if u2i (chessboard.n_files) < j1 then
nil_move
else
let
val v = chessboard_vt_get (chessboard, i1, j1) : int
in
if v <> EMPTY_SQUARE then
nil_move
else
@(i1, j1)
end
end

val move0 = try_move (chessboard, 1, 2)
val move1 = try_move (chessboard, 2, 1)
val move2 = try_move (chessboard, 1, ~2)
val move3 = try_move (chessboard, 2, ~1)
val move4 = try_move (chessboard, ~1, 2)
val move5 = try_move (chessboard, ~2, 1)
val move6 = try_move (chessboard, ~1, ~2)
val move7 = try_move (chessboard, ~2, ~1)
in
@(move0, move1, move2, move3, move4, move5, move6, move7)
end

fn
count_following_moves
{n_ranks, n_files : pos}
{i, j       : int}
{n_position : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
move       : move_t (i, j),
n_position : int n_position) :
uint =
if move_t_is_nil move then
0U
else
let
fn
succ_if_move_is_not_nil
{i, j : int}
(w    : uint,
move : move_t (i, j)) :<>
uint =
if move_t_is_nil move then
w
else
succ w

val @(i, j) = move
val _ = chessboard_vt_set<int> (chessboard, i, j,
succ n_position)
val following_moves = possible_moves (chessboard, i, j)

val w = 0U
val w = succ_if_move_is_not_nil (w, following_moves.0)
val w = succ_if_move_is_not_nil (w, following_moves.1)
val w = succ_if_move_is_not_nil (w, following_moves.2)
val w = succ_if_move_is_not_nil (w, following_moves.3)
val w = succ_if_move_is_not_nil (w, following_moves.4)
val w = succ_if_move_is_not_nil (w, following_moves.5)
val w = succ_if_move_is_not_nil (w, following_moves.6)
val w = succ_if_move_is_not_nil (w, following_moves.7)

val _ = chessboard_vt_set<int> (chessboard, i, j, EMPTY_SQUARE)
in
w
end

fn
pick_w (w0 : uint,
w1 : uint,
w2 : uint,
w3 : uint,
w4 : uint,
w5 : uint,
w6 : uint,
w7 : uint) :<>
uint =
let
fn
next_pick (u : uint,
v : uint) :<>
uint =
if v = 0U then
u
else if u = 0U then
v
else
min (u, v)

val w = 0U
val w = next_pick (w, w0)
val w = next_pick (w, w1)
val w = next_pick (w, w2)
val w = next_pick (w, w3)
val w = next_pick (w, w4)
val w = next_pick (w, w5)
val w = next_pick (w, w6)
val w = next_pick (w, w7)
in
w
end

fn
next_moves {n_ranks, n_files : pos}
{i, j       : int}
{n_position : int}
(chessboard : !chessboard_vt (int, n_ranks, n_files),
i          : int i,
j          : int j,
n_position : int n_position) :
knights_moves_t =
(* Prune and sort the moves according to Warnsdorff’s heuristic,
keeping only moves that have the minimum number of legal
following moves. *)
let
val moves = possible_moves (chessboard, i, j)
val w0 = count_following_moves (chessboard, moves.0, n_position)
val w1 = count_following_moves (chessboard, moves.1, n_position)
val w2 = count_following_moves (chessboard, moves.2, n_position)
val w3 = count_following_moves (chessboard, moves.3, n_position)
val w4 = count_following_moves (chessboard, moves.4, n_position)
val w5 = count_following_moves (chessboard, moves.5, n_position)
val w6 = count_following_moves (chessboard, moves.6, n_position)
val w7 = count_following_moves (chessboard, moves.7, n_position)
val w = pick_w (w0, w1, w2, w3, w4, w5, w6, w7)
in
if w = 0U then
@(nil_move, nil_move, nil_move, nil_move,
nil_move, nil_move, nil_move, nil_move)
else
@(if w0 = w then moves.0 else nil_move,
if w1 = w then moves.1 else nil_move,
if w2 = w then moves.2 else nil_move,
if w3 = w then moves.3 else nil_move,
if w4 = w then moves.4 else nil_move,
if w5 = w then moves.5 else nil_move,
if w6 = w then moves.6 else nil_move,
if w7 = w then moves.7 else nil_move)
end

fn
make_and_fprint_tours
{n_ranks, n_files : int}
{i, j        : int}
{max_tours   : int}
(f           : FILEref,
n_ranks     : int n_ranks,
n_files     : int n_files,
i           : int i,
j           : int j,
max_tours   : int max_tours,
closed_only : bool) :
void =
{
val n_ranks = max (1, n_ranks)
val n_files = max (1, n_files)
val i = max (1, min (n_ranks, i))
val j = max (1, min (n_files, j))
val max_tours = max (1, max_tours)

val n_ranks = i2u n_ranks
val n_files = i2u n_files

val i_start = i
val j_start = j

var tours_printed : int = 0

val chessboard =
chessboard_vt_make<int> (n_ranks, n_files, g1ofg0 EMPTY_SQUARE)

fun
explore {n_ranks, n_files : pos}
{i, j          : int}
{n_position    : int}
(chessboard    : !chessboard_vt (int, n_ranks, n_files),
i             : int i,
j             : int j,
n_position    : int n_position,
tours_printed : &int) :
void =
if tours_printed < max_tours then
let
fn
print_board {i1, j1 : int}
(chessboard    : !chessboard_vt (int, n_ranks,
n_files),
tours_printed : &int) :
void =
begin
tours_printed := succ tours_printed;
fprintln! (f, "Tour number ", tours_printed);
knights_tour_moves_fprint (f, chessboard);
fprintln! (f);
knights_tour_board_fprint (f, chessboard);
fprintln! (f);
fprintln! (f)
end

fn
satisfies_closedness
{i1, j1 : int}
(move : move_t (i1, j1)) :
bool =
if closed_only then
let
val @(i1, j1) = move
val i_diff = abs (i1 - i_start)
val j_diff = abs (j1 - j_start)
in
(i_diff = 1 && j_diff = 2)
|| (i_diff = 2 && j_diff = 1)
end
else
true

fn
try_last_move
{i1, j1 : int}
(chessboard    : !chessboard_vt (int, n_ranks,
n_files),
move          : move_t (i1, j1),
tours_printed : &int) :
void =
if ~move_t_is_nil move && satisfies_closedness move then
let
val @(i1, j1) = move
in
chessboard_vt_set<int> (chessboard, i1, j1,
n_position + 1);
print_board (chessboard, tours_printed);
chessboard_vt_set<int> (chessboard, i1, j1,
EMPTY_SQUARE)
end

fun
explore_inner (chessboard : !chessboard_vt (int, n_ranks,
n_files),
tours_printed : &int) :
void =
if u2i (chessboard.n_squares) - n_position = 1 then
(* Is the last move possible? If so, make it and print
the board. (Only zero or one of the moves can be
non-nil.) *)
let
val moves = possible_moves (chessboard, i, j)
in
try_last_move (chessboard, moves.0, tours_printed);
try_last_move (chessboard, moves.1, tours_printed);
try_last_move (chessboard, moves.2, tours_printed);
try_last_move (chessboard, moves.3, tours_printed);
try_last_move (chessboard, moves.4, tours_printed);
try_last_move (chessboard, moves.5, tours_printed);
try_last_move (chessboard, moves.6, tours_printed);
try_last_move (chessboard, moves.7, tours_printed)
end
else
let
val moves = next_moves (chessboard, i, j, n_position)
macdef explore_move (move) =
begin
if ~move_t_is_nil ,(move) then
explore (chessboard, (,(move)).0, (,(move)).1,
succ n_position, tours_printed)
end
in
explore_move (moves.0);
explore_move (moves.1);
explore_move (moves.2);
explore_move (moves.3);
explore_move (moves.4);
explore_move (moves.5);
explore_move (moves.6);
explore_move (moves.7)
end
in
chessboard_vt_set<int> (chessboard, i, j, n_position);
explore_inner (chessboard, tours_printed);
chessboard_vt_set<int> (chessboard, i, j, EMPTY_SQUARE)
end

val _ = explore (chessboard, i, j, 1, tours_printed)

val _ = \$UNSAFE.castvwtp0{void} chessboard
}

fn
algebraic_notation_to_move (s : string) :
move_t =
let
val s = g1ofg0 s
val n = string_length s
in
if n = 2 then
let
val i = g1ofg0 (char2i (s[1]) - char2i ('0'))
val j = g1ofg0 (char2i (s[0]) - char2i ('a') + 1)
in
@(i, j)
end
else
@(1, 1)
end

implement
main0 (argc, argv) =
{
val @(i, j) =
begin
if 2 <= argc then
algebraic_notation_to_move (argv[1])
else
@(1, 1)
end : move_t

val max_tours =
begin
if 3 <= argc then
\$extfcall (int, "atoi", argv[2])
else
1
end : int
val max_tours = g1ofg0 max_tours

val closed_only =
begin
if 4 <= argc then
argv[3] = "closed"
else
false
end : bool

val _ = make_and_fprint_tours (stdout_ref, 8, 8, i, j, max_tours,
closed_only)
}```
Output:

\$ ./knights_tour c5 2 closed

```Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 58 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 63 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 60 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h

Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 60 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 61 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 58 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h
```

## AutoHotkey

Library: GDIP
```#SingleInstance, Force
#NoEnv
SetBatchLines, -1
; Uncomment if Gdip.ahk is not in your standard library
;#Include, Gdip.ahk
If !pToken := Gdip_Startup(){
MsgBox, 48, Gdiplus error!, Gdiplus failed to start. Please ensure you have Gdiplus on your system.
ExitApp
}
; I've added a simple new function here, just to ensure if anyone is having any problems then to make sure they are using the correct library version
if (Gdip_LibraryVersion() < 1.30)
{
ExitApp
}
OnExit, Exit
tour := "a1 b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 "
; Knight's tour with maximum symmetry by George Jelliss, http://www.mayhematics.com/t/8f.htm
; I know, I know, but I followed the task outline to the letter! Besides, this path is the prettiest.

; Input: starting square
InputBox, start, Knight's Tour Start, Enter Knight's starting location in algebraic notation:, , , , , , , , b3
i := InStr(tour, start)
If i=0
{
Msgbox Error, please try again.
}
; Output: move sequence
Msgbox % tour := SubStr(tour, i) . SubStr(tour, 1, i-1)

; Animation
tour .= SubStr(tour, 1, 3)
, CellSize := 30 ; pixels
, Width := Height := 9*CellSize
, TopLeftX := (A_ScreenWidth - Width) // 2
, TopLeftY := (A_ScreenHeight - Height) // 2
Gui, -Caption +E0x80000 +LastFound +AlwaysOnTop +ToolWindow +OwnDialogs
Gui, Show, NA ; show board (currently transparent)
hwnd1 := WinExist() ; required for Gdip
OnMessage(0x201, "WM_LBUTTONDOWN")
, hbm := CreateDIBSection(Width, Height)
, hdc := CreateCompatibleDC()
, obm := SelectObject(hdc, hbm)
, G := Gdip_GraphicsFromHDC(hdc)
, Gdip_SetSmoothingMode(G, 4)

Loop 1 ; remove '1' and uncomment next line to loop infinitely
{
;Gdip_GraphicsClear(G) ; uncomment to loop infinitely
cOdd := "0xFFFFCE9E" ; create brushes
, cEven := "0xFFD18B47"
, pBrushOdd := Gdip_BrushCreateSolid(cOdd)
, pBrushEven := Gdip_BrushCreateSolid(cEven)

Loop 64 ; layout board
{
Row := mod(A_Index-1,8)+1
, Col := (A_Index-1)//8+1
, Gdip_FillRectangle(G, mod(Row+Col,2) ? pBrushOdd : pBrushEven, Col * CellSize + 1, Row * CellSize + 1, CellSize - 2, CellSize - 2)
}
Gdip_DeleteBrush(pBrushOdd) ; cleanup memory
, Gdip_DeleteBrush(pBrushEven)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board

, pPen := Gdip_CreatePen(0x66FF0000, CellSize/10) ; create pen
, Algebraic := SubStr(tour,1,2) ; get starting coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize

Loop 64 ; trace path
{
Sleep, 0.5*1000
xold := x, yold := y ; a line has start and end points
, Algebraic := SubStr(tour,(A_Index)*3+1,2) ; get new coordinates
, x := (Asc(SubStr(Algebraic, 1, 1))-96+0.5)*CellSize
, y := (9.5-SubStr(Algebraic, 2, 1))*CellSize
, Gdip_DrawLine(G, pPen, xold, yold, x, y)
, UpdateLayeredWindow(hwnd1, hdc, TopLeftX, TopLeftY, Width, Height) ; update board
}
Gdip_DeletePen(pPen)
}
Return

GuiEscape:
ExitApp

Exit:
Gdip_Shutdown(pToken)
ExitApp

WM_LBUTTONDOWN()
{
If (A_Gui = 1)
PostMessage, 0xA1, 2
}
```
Output:

For start at b3

`b3 d2 c4 a5 b7 d8 e6 d4 b5 c7 a8 b6 c8 a7 c6 b8 a6 b4 d5 e3 d1 b2 a4 c5 d7 f8 h7 f6 g8 h6 f7 h8 g6 e7 f5 h4 g2 e1 d3 e5 g4 f2 h1 g3 f1 h2 f3 g1 h3 g5 e4 d6 e8 g7 h5 f4 e2 c1 a2 c3 b1 a3 c2 a1 `

... plus an animation.

## AWK

```# syntax: GAWK -f KNIGHTS_TOUR.AWK [-v sr=x] [-v sc=x]
#
# examples:
#   GAWK -f KNIGHTS_TOUR.AWK                   (default)
#   GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=1   start at top left (default)
#   GAWK -f KNIGHTS_TOUR.AWK -v sr=1 -v sc=8   start at top right
#   GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=8   start at bottom right
#   GAWK -f KNIGHTS_TOUR.AWK -v sr=8 -v sc=1   start at bottom left
#
BEGIN {
N = 8 # board size
if (sr == "") { sr = 1 } # starting row
if (sc == "") { sc = 1 } # starting column
split("2  2 -2 -2 1  1 -1 -1",X," ")
split("1 -1  1 -1 2 -2  2 -2",Y," ")
printf("\n%dx%d board: starting row=%d col=%d\n",N,N,sr,sc)
move(sr,sc,0)
exit(1)
}
function move(x,y,m) {
if (cantMove(x,y)) {
return(0)
}
P[x,y] = ++m
if (m == N ^ 2) {
printBoard()
exit(0)
}
tryBestMove(x,y,m)
}
function cantMove(x,y) {
return( P[x,y] || x<1 || x>N || y<1 || y>N )
}
function tryBestMove(x,y,m,  i) {
i = bestMove(x,y)
move(x+X[i],y+Y[i],m)
}
function bestMove(x,y,  arg1,arg2,c,i,min,out) {
# Warnsdorff's rule: go to where there are fewest next moves
min = N ^ 2 + 1
for (i in X) {
arg1 = x + X[i]
arg2 = y + Y[i]
if (!cantMove(arg1,arg2)) {
c = countNext(arg1,arg2)
if (c < min) {
min = c
out = i
}
}
}
return(out)
}
function countNext(x,y,  i,out) {
for (i in X) {
out += (!cantMove(x+X[i],y+Y[i]))
}
return(out)
}
function printBoard(  i,j,leng) {
leng = length(N*N)
for (i=1; i<=N; i++) {
for (j=1; j<=N; j++) {
printf(" %*d",leng,P[i,j])
}
printf("\n")
}
}
```

output:

```8x8 board: starting row=1 col=1
1 50 15 32 61 28 13 30
16 33 64 55 14 31 60 27
51  2 49 44 57 62 29 12
34 17 56 63 54 47 26 59
3 52 45 48 43 58 11 40
18 35 20 53 46 41  8 25
21  4 37 42 23  6 39 10
36 19 22  5 38  9 24  7
```

## BASIC

### ANSI BASIC

Translation of: BBC BASIC
Works with: Decimal BASIC

ANSI BASIC does not allow function parameters to be passed by reference, so X and Y were made global variables.

```100 DECLARE EXTERNAL FUNCTION choosemove
110 !
120 RANDOMIZE
130 PUBLIC NUMERIC X, Y, TRUE, FALSE
140 LET TRUE = -1
150 LET FALSE = 0
160 !
170 SET WINDOW 1,512,1,512
180 SET AREA COLOR "black"
190 FOR x=0 TO 512-128 STEP 128
200    FOR y=0 TO 512-128 STEP 128
210       PLOT AREA:x+64,y;x+128,y;x+128,y+64;x+64,y+64
220       PLOT AREA:x,y+64;x+64,y+64;x+64,y+128;x,y+128
230    NEXT y
240 NEXT x
250 !
260 SET LINE COLOR "red"
270 SET LINE WIDTH 6
280 !
290 PUBLIC NUMERIC Board(0 TO 7,0 TO 7)
300 LET X = 0
310 LET Y = 0
320 LET Total = 0
330 DO
340    LET Board(X,Y) = TRUE
350    PLOT LINES: X*64+32,Y*64+32;
360    LET Total = Total + 1
370 LOOP UNTIL choosemove(X, Y) = FALSE
380 IF Total <> 64 THEN STOP
390 END
400 !
410 EXTERNAL FUNCTION choosemove(X1, Y1)
420 DECLARE EXTERNAL SUB trymove
430 LET M = 9
440 CALL trymove(X1+1, Y1+2, M, newx, newy)
450 CALL trymove(X1+1, Y1-2, M, newx, newy)
460 CALL trymove(X1-1, Y1+2, M, newx, newy)
470 CALL trymove(X1-1, Y1-2, M, newx, newy)
480 CALL trymove(X1+2, Y1+1, M, newx, newy)
490 CALL trymove(X1+2, Y1-1, M, newx, newy)
500 CALL trymove(X1-2, Y1+1, M, newx, newy)
510 CALL trymove(X1-2, Y1-1, M, newx, newy)
520 IF M=9 THEN
530    LET choosemove = FALSE
540    EXIT FUNCTION
550 END IF
560 LET X = newx
570 LET Y = newy
580 LET choosemove = TRUE
590 END FUNCTION
600 !
610 EXTERNAL SUB trymove(X, Y, M, newx, newy)
620 !
630 DECLARE EXTERNAL FUNCTION validmove
640 IF validmove(X,Y) = 0 THEN EXIT SUB
650 IF validmove(X+1,Y+2) <> 0 THEN LET N = N + 1
660 IF validmove(X+1,Y-2) <> 0 THEN LET N = N + 1
670 IF validmove(X-1,Y+2) <> 0 THEN LET N = N + 1
680 IF validmove(X-1,Y-2) <> 0 THEN LET N = N + 1
690 IF validmove(X+2,Y+1) <> 0 THEN LET N = N + 1
700 IF validmove(X+2,Y-1) <> 0 THEN LET N = N + 1
710 IF validmove(X-2,Y+1) <> 0 THEN LET N = N + 1
720 IF validmove(X-2,Y-1) <> 0 THEN LET N = N + 1
730 IF N>M THEN EXIT SUB
740 IF N=M AND RND<.5 THEN EXIT SUB
750 LET M = N
760 LET newx = X
770 LET newy = Y
780 END SUB
790 !
800 EXTERNAL FUNCTION validmove(X,Y)
810 LET validmove = FALSE
820 IF X<0 OR X>7 OR Y<0 OR Y>7 THEN EXIT FUNCTION
830 IF Board(X,Y)=FALSE THEN LET validmove = TRUE
840 END FUNCTION
```

### BBC BASIC

```      VDU 23,22,256;256;16,16,16,128
VDU 23,23,4;0;0;0;
OFF
GCOL 4,15
FOR x% = 0 TO 512-128 STEP 128
RECTANGLE FILL x%,0,64,512
NEXT
FOR y% = 0 TO 512-128 STEP 128
RECTANGLE FILL 0,y%,512,64
NEXT
GCOL 9

DIM Board%(7,7)
X% = 0
Y% = 0
Total% = 0
REPEAT
Board%(X%,Y%) = TRUE
IF Total% DRAW X%*64+32,Y%*64+32 ELSE MOVE X%*64+32,Y%*64+32
Total% += 1
UNTIL NOT FNchoosemove(X%, Y%)
IF Total%<>64 STOP
REPEAT WAIT 1 : UNTIL FALSE
END

DEF FNchoosemove(RETURN X%, RETURN Y%)
LOCAL M%, newx%, newy%
M% = 9
PROCtrymove(X%+1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%+1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%+2, M%, newx%, newy%)
PROCtrymove(X%-1, Y%-2, M%, newx%, newy%)
PROCtrymove(X%+2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%+2, Y%-1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%+1, M%, newx%, newy%)
PROCtrymove(X%-2, Y%-1, M%, newx%, newy%)
IF M%=9 THEN = FALSE
X% = newx% : Y% = newy%
= TRUE

DEF PROCtrymove(X%, Y%, RETURN M%, RETURN newx%, RETURN newy%)
LOCAL N%
IF NOT FNvalidmove(X%,Y%) THEN ENDPROC
IF FNvalidmove(X%+1,Y%+2) N% += 1
IF FNvalidmove(X%+1,Y%-2) N% += 1
IF FNvalidmove(X%-1,Y%+2) N% += 1
IF FNvalidmove(X%-1,Y%-2) N% += 1
IF FNvalidmove(X%+2,Y%+1) N% += 1
IF FNvalidmove(X%+2,Y%-1) N% += 1
IF FNvalidmove(X%-2,Y%+1) N% += 1
IF FNvalidmove(X%-2,Y%-1) N% += 1
IF N%>M% THEN ENDPROC
IF N%=M% IF RND(2)=1 THEN ENDPROC
M% = N%
newx% = X% : newy% = Y%
ENDPROC

DEF FNvalidmove(X%,Y%)
IF X%<0 OR X%>7 OR Y%<0 OR Y%>7 THEN = FALSE
= NOT(Board%(X%,Y%))
```

## Bracmat

```  ( knightsTour
=     validmoves WarnsdorffSort algebraicNotation init solve
, x y fieldsToVisit
.   ~
|   ( validmoves
=   x y jumps moves
.   !arg:(?x.?y)
& :?moves
& ( jumps
=   dx dy Fs fxs fys fx fy
.   !arg:(?dx.?dy)
& 1 -1:?Fs
& !Fs:?fxs
&   whl
' ( !fxs:%?fx ?fxs
& !Fs:?fys
&   whl
' ( !fys:%?fy ?fys
&     (   (!x+!fx*!dx.!y+!fy*!dy)
: (>0:<9.>0:<9)
|
)
!moves
: ?moves
)
)
)
& jumps\$(1.2)
& jumps\$(2.1)
& !moves
)
& ( init
=   fields x y
.   :?fields
& 0:?x
&   whl
' ( 1+!x:<9:?x
& 0:?y
&   whl
' ( 1+!y:<9:?y
& (!x.!y) !fields:?fields
)
)
& !fields
)
& init\$:?fieldsToVisit
& ( WarnsdorffSort
=   sum moves elm weightedTerms
.   ( weightedTerms
=   pos alts fieldsToVisit moves move weight
.     !arg:(%?pos ?alts.?fieldsToVisit)
&   (   !fieldsToVisit:!pos
& (0.!pos)
|   !fieldsToVisit:? !pos ?
& validmoves\$!pos:?moves
& 0:?weight
&   whl
' ( !moves:%?move ?moves
& (   !fieldsToVisit:? !move ?
& !weight+1:?weight
|
)
)
& (!weight.!pos)
| 0
)
+ weightedTerms\$(!alts.!fieldsToVisit)
| 0
)
& weightedTerms\$!arg:?sum
& :?moves
&   whl
' ( !sum:(#.?elm)+?sum
& !moves !elm:?moves
)
& !moves
)
& ( solve
=   pos alts fieldsToVisit A Z tailOfSolution
.   !arg:(%?pos ?alts.?fieldsToVisit)
&   (   !fieldsToVisit:?A !pos ?Z
& ( !A !Z:&
|   solve
\$ ( WarnsdorffSort\$(validmoves\$!pos.!A !Z)
. !A !Z
)
)
| solve\$(!alts.!fieldsToVisit)
)
: ?tailOfSolution
& !pos !tailOfSolution
)
& ( algebraicNotation
=   x y
.     !arg:(?x.?y) ?arg
&   str\$(chr\$(asc\$a+!x+-1) !y " ")
algebraicNotation\$!arg
|
)
& @(!arg:?x #?y)
& asc\$!x+-1*asc\$a+1:?x
&   str
\$ (algebraicNotation\$(solve\$((!x.!y).!fieldsToVisit)))
)
& out\$(knightsTour\$a1);```
`a1 b3 a5 b7 d8 f7 h8 g6 f8 h7 g5 h3 g1 e2 c1 a2 b4 a6 b8 c6 a7 c8 e7 g8 h6 g4 h2 f1 d2 b1 a3 c2 e1 f3 h4 g2 e3 d1 b2 a4 c3 b5 d4 f5 d6 c4 e5 d3 f2 h1 g3 e4 c5 d7 b6 a8 c7 d5 f4 e6 g7 e8 f6 h5`

## C

For an animated version using OpenGL, see Knight's tour/C.

The following draws on console the progress of the horsie. Specify board size on commandline, or use default 8.

```#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <unistd.h>

typedef unsigned char cell;
int dx[] = { -2, -2, -1, 1, 2,  2,  1, -1 };
int dy[] = { -1,  1,  2, 2, 1, -1, -2, -2 };

void init_board(int w, int h, cell **a, cell **b)
{
int i, j, k, x, y, p = w + 4, q = h + 4;
/* b is board; a is board with 2 rows padded at each side */
a[0] = (cell*)(a + q);
b[0] = a[0] + 2;

for (i = 1; i < q; i++) {
a[i] = a[i-1] + p;
b[i] = a[i] + 2;
}

memset(a[0], 255, p * q);
for (i = 0; i < h; i++) {
for (j = 0; j < w; j++) {
for (k = 0; k < 8; k++) {
x = j + dx[k], y = i + dy[k];
if (b[i+2][j] == 255) b[i+2][j] = 0;
b[i+2][j] += x >= 0 && x < w && y >= 0 && y < h;
}
}
}
}

#define E "\033["
int walk_board(int w, int h, int x, int y, cell **b)
{
int i, nx, ny, least;
int steps = 0;
printf(E"H"E"J"E"%d;%dH"E"32m[]"E"m", y + 1, 1 + 2 * x);

while (1) {
/* occupy cell */
b[y][x] = 255;

/* reduce all neighbors' neighbor count */
for (i = 0; i < 8; i++)
b[ y + dy[i] ][ x + dx[i] ]--;

/* find neighbor with lowest neighbor count */
least = 255;
for (i = 0; i < 8; i++) {
if (b[ y + dy[i] ][ x + dx[i] ] < least) {
nx = x + dx[i];
ny = y + dy[i];
least = b[ny][nx];
}
}

if (least > 7) {
printf(E"%dH", h + 2);
return steps == w * h - 1;
}

if (steps++) printf(E"%d;%dH[]", y + 1, 1 + 2 * x);
x = nx, y = ny;
printf(E"%d;%dH"E"31m[]"E"m", y + 1, 1 + 2 * x);
fflush(stdout);
usleep(120000);
}
}

int solve(int w, int h)
{
int x = 0, y = 0;
cell **a, **b;
a = malloc((w + 4) * (h + 4) + sizeof(cell*) * (h + 4));
b = malloc((h + 4) * sizeof(cell*));

while (1) {
init_board(w, h, a, b);
if (walk_board(w, h, x, y, b + 2)) {
printf("Success!\n");
return 1;
}
if (++x >= w) x = 0, y++;
if (y >= h) {
printf("Failed to find a solution\n");
return 0;
}
printf("Any key to try next start position");
getchar();
}
}

int main(int c, char **v)
{
int w, h;
if (c < 2 || (w = atoi(v[1])) <= 0) w = 8;
if (c < 3 || (h = atoi(v[2])) <= 0) h = w;
solve(w, h);

return 0;
}
```

## C#

```using System;
using System.Collections.Generic;

namespace prog
{
class MainClass
{
const int N = 8;

readonly static int[,] moves = { {+1,-2},{+2,-1},{+2,+1},{+1,+2},
{-1,+2},{-2,+1},{-2,-1},{-1,-2} };
struct ListMoves
{
public int x, y;
public ListMoves( int _x, int _y ) { x = _x; y = _y; }
}

public static void Main (string[] args)
{
int[,] board = new int[N,N];
board.Initialize();

int x = 0,						// starting position
y = 0;

List<ListMoves> list = new List<ListMoves>(N*N);
list.Add( new ListMoves(x,y) );

do
{
if ( Move_Possible( board, x, y ) )
{
int move = board[x,y];
board[x,y]++;
x += moves[move,0];
y += moves[move,1];
list.Add( new ListMoves(x,y) );
}
else
{
if ( board[x,y] >= 8 )
{
board[x,y] = 0;
list.RemoveAt(list.Count-1);
if ( list.Count == 0 )
{
Console.WriteLine( "No solution found." );
return;
}
x = list[list.Count-1].x;
y = list[list.Count-1].y;
}
board[x,y]++;
}
}
while( list.Count < N*N );

int last_x = list[0].x,
last_y = list[0].y;
string letters = "ABCDEFGH";
for( int i=1; i<list.Count; i++ )
{
Console.WriteLine( string.Format("{0,2}:  ", i) + letters[last_x] + (last_y+1) + " - " + letters[list[i].x] + (list[i].y+1) );

last_x = list[i].x;
last_y = list[i].y;
}
}

static bool Move_Possible( int[,] board, int cur_x, int cur_y )
{
if ( board[cur_x,cur_y] >= 8 )
return false;

int new_x = cur_x + moves[board[cur_x,cur_y],0],
new_y = cur_y + moves[board[cur_x,cur_y],1];

if ( new_x >= 0 && new_x < N && new_y >= 0 && new_y < N && board[new_x,new_y] == 0 )
return true;

return false;
}
}
}
```

## C++

Works with: C++11

Uses Warnsdorff's rule and (iterative) backtracking if that fails.

```#include <iostream>
#include <iomanip>
#include <array>
#include <string>
#include <tuple>
#include <algorithm>
using namespace std;

template<int N = 8>
class Board
{
public:
array<pair<int, int>, 8> moves;
array<array<int, N>, N> data;

Board()
{
moves[0] = make_pair(2, 1);
moves[1] = make_pair(1, 2);
moves[2] = make_pair(-1, 2);
moves[3] = make_pair(-2, 1);
moves[4] = make_pair(-2, -1);
moves[5] = make_pair(-1, -2);
moves[6] = make_pair(1, -2);
moves[7] = make_pair(2, -1);
}

array<int, 8> sortMoves(int x, int y) const
{
array<tuple<int, int>, 8> counts;
for(int i = 0; i < 8; ++i)
{
int dx = get<0>(moves[i]);
int dy = get<1>(moves[i]);

int c = 0;
for(int j = 0; j < 8; ++j)
{
int x2 = x + dx + get<0>(moves[j]);
int y2 = y + dy + get<1>(moves[j]);

if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N)
continue;
if(data[y2][x2] != 0)
continue;

c++;
}

counts[i] = make_tuple(c, i);
}

// Shuffle to randomly break ties
random_shuffle(counts.begin(), counts.end());

// Lexicographic sort
sort(counts.begin(), counts.end());

array<int, 8> out;
for(int i = 0; i < 8; ++i)
out[i] = get<1>(counts[i]);
return out;
}

void solve(string start)
{
for(int v = 0; v < N; ++v)
for(int u = 0; u < N; ++u)
data[v][u] = 0;

int x0 = start[0] - 'a';
int y0 = N - (start[1] - '0');
data[y0][x0] = 1;

array<tuple<int, int, int, array<int, 8>>, N*N> order;
order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0));

int n = 0;
while(n < N*N-1)
{
int x = get<0>(order[n]);
int y = get<1>(order[n]);

bool ok = false;
for(int i = get<2>(order[n]); i < 8; ++i)
{
int dx = moves[get<3>(order[n])[i]].first;
int dy = moves[get<3>(order[n])[i]].second;

if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N)
continue;
if(data[y + dy][x + dx] != 0)
continue;

get<2>(order[n]) = i + 1;
++n;
data[y+dy][x+dx] = n + 1;
order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy));
ok = true;
break;
}

if(!ok) // Failed. Backtrack.
{
data[y][x] = 0;
--n;
}
}
}

template<int N>
friend ostream& operator<<(ostream &out, const Board<N> &b);
};

template<int N>
ostream& operator<<(ostream &out, const Board<N> &b)
{
for (int v = 0; v < N; ++v)
{
for (int u = 0; u < N; ++u)
{
if (u != 0) out << ",";
out << setw(3) << b.data[v][u];
}
out << endl;
}
return out;
}

int main()
{
Board<5> b1;
b1.solve("c3");
cout << b1 << endl;

Board<8> b2;
b2.solve("b5");
cout << b2 << endl;

Board<31> b3; // Max size for <1000 squares
b3.solve("a1");
cout << b3 << endl;
return 0;
}
```

Output:

``` 23, 16, 11,  6, 21
10,  5, 22, 17, 12
15, 24,  1, 20,  7
4,  9, 18, 13,  2
25, 14,  3,  8, 19

63, 20,  3, 24, 59, 36,  5, 26
2, 23, 64, 37,  4, 25, 58, 35
19, 62, 21, 50, 55, 60, 27,  6
22,  1, 54, 61, 38, 45, 34, 57
53, 18, 49, 44, 51, 56,  7, 28
12, 15, 52, 39, 46, 31, 42, 33
17, 48, 13, 10, 43, 40, 29,  8
14, 11, 16, 47, 30,  9, 32, 41

275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33
18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198
111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869
114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130
271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35
16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194
109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201
282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132
267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37
14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190
107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205
286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134
263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39
12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186
105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209
290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136
259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41
10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182
103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213
294,  9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138
255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43
8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178
101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165
298,  7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140
251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45
6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168
99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143
240,  5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82
63,  2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47
4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84
1, 62,  3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145
```

## Common Lisp

Works with: clisp version 2.49

This interactive program will ask for a starting case in algebraic notation and, also, whether a closed tour is desired. Each next move is selected according to Warnsdorff's rule; ties are broken at random.

The closed tour algorithm is quite crude: just find tours over and over until one happens to be closed by chance.

This code is quite verbose: I tried to make it easy for myself and for others to follow and understand. I'm not a Lisp expert, so I probably missed some idiomatic shortcuts I could have used to make it shorter.

For some reason, the interactive part does not work with SBCL, but it works fine with CLISP.

```;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;   Solving the knight's tour.                     ;;;
;;;   Warnsdorff's rule with random tie break.       ;;;
;;;   Optionally outputs a closed tour.              ;;;
;;;   Options from interactive prompt.               ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defparameter *side* 8)

(defun generate-chessboard (n)
(loop for i below n append
(loop for j below n collect (complex i j))))

(defparameter *chessboard*
(generate-chessboard *side*))

(defun complex->algebraic (n)
;; returns a string like "b2"
(concatenate 'string
;; 'a' is char #97: add it to the offset
(string (character  (+ 97 (realpart n))))
;; indices start at 0, but algebraic starts at 1
(string (digit-char (+ 1  (imagpart n))))))

(defun algebraic->complex (string)
;; takes a string like "e4"
(let ((row (char string 0))
(col (char string 1)))
(complex (- (char-code row) 97)
(- (digit-char-p col) 1))))

(defconstant *knight-directions*
(list
(complex  1  2)
(complex  2  1)
(complex  1 -2)
(complex  2 -1)
(complex -1  2)
(complex -2  1)
(complex -1 -2)
(complex -2 -1)))

(defun find-legal-moves (moves-list)
;; 2. the move must not be on a case already visited
(remove-if (lambda (m) (member m moves-list))
;; 1. the move must be within the chessboard
(intersection
(mapcar (lambda (i) (+ (car moves-list) i)) *knight-directions*)
*chessboard*)))

;; Select between two moves by Warnsdorff's rule:
;; pick the one with the lowest index or else
;; randomly break the tie.
;; Takes a cons in the form (n . #C(x y)).
;; This will be the sorting rule for picking the next move.
(defun w-rule (a b)
(cond ((< (car a) (car b)) t)
((> (car a) (car b)) nil)
((= (car a) (car b))
(zerop (random 2)))))

;; For every legal move in a given position,
;; look forward one move and return a cons
;; in the form (n . #C(x y)) where n is
;; how many next free moves follow the first move.
(defun return-weighted-moves (moves)
(let ((candidates (find-legal-moves moves)))
(loop for mv in candidates collect
(cons
(list-length (find-legal-moves (cons mv moves)))
mv))))

;; Given a list of weighted moves (as above),
;; pick one according to the w-rule
(defun pick-among-weighted-moves (moves)
;; prune dead ends one move early
(let ((possible-moves
(remove-if (lambda(m) (zerop (car m))) moves)))
(cdar (sort possible-moves #'w-rule))))

(defun make-move (moves-list)
(let ((next-move
(if (< (list-length moves-list) (1- (list-length *chessboard*)))
(pick-among-weighted-moves (return-weighted-moves moves-list))
(car (find-legal-moves moves-list)))))
(cons next-move moves-list)))

(defun make-tour (moves-list)
;; takes a list of moves as an argument
(if (null (car moves-list)) ; last move not found: start over
(make-tour (last moves-list))
(if (= (list-length moves-list) (list-length *chessboard*))
moves-list
(make-tour (make-move moves-list)))))

(defun make-closed-tour (moves-list)
(let ((tour (make-tour moves-list)))
(if (tour-closed-p tour)
tour
(make-closed-tour moves-list))))

(defun tour-closed-p (tour)
;; takes a full tour as an argument
(let ((start (car (last tour)))
(end (car tour)))
;; is the first position a legal move, when
;; viewed from the last move?
(if (member start (find-legal-moves (list end))) ; find-legal-moves takes a list
t nil)))

(defun print-tour-linear (tour)
;; takes a tour (moves list) with the last move first
;; and prints it nicely in algebraic notation
(let ((moves (mapcar #'complex->algebraic (reverse tour))))
(format t "~{~A~^ -> ~}" moves)))

(defun tour->matrix (tour)
;; takes a tour and makes a row-by-row 2D matrix
;; from top to bottom (for further formatting & printing)
(flet ((index-tour (tour) ; 1st local function
(loop for i below (length tour)
;; starting from index 1, not 0, so add 1;
;; reverse because the last move is still in the car
collect (cons (nth i (reverse tour)) (1+ i))))
(get-row (n tour)  ; 2nd local function
;; in every row, the imaginary part (vertical offset) stays the same
(remove-if-not (lambda (e) (= n (imagpart (car e)))) tour)))
(let* ((indexed-tour (index-tour tour))
(ordered-indexed-tour
;; make a list of ordered rows
(loop for i from (1- *side*) downto 0 collect
(sort (get-row i indexed-tour)
(lambda (a b) (< (realpart (car a)) (realpart (car b))))))))
;; clean up, leaving only the indices
(mapcar (lambda (e) (mapcar #'cdr e)) ordered-indexed-tour))))

(defun print-tour-matrix (tour)
(mapcar (lambda (row)
(format t "~{~3d~}~&" row)) (tour->matrix tour)))

;;; Handling options

(defstruct options
closed
start
grid)

(defparameter *opts* (make-options))

;;; Interactive part

(defun prompt()
(format t "Starting case (leave blank for random)? ")
(let ((start (string (read-line))))
(if (member start (mapcar #'complex->algebraic *chessboard*) :test #'equal)
(setf (options-start *opts*) start))
(format t "Require a closed tour (yes or default to no)? ")
(if (or (equal closed "y") (equal closed "yes"))
(setf (options-closed *opts*) t)))))

(defun main ()
(let* ((start
(if (options-start *opts*)
(algebraic->complex (options-start *opts*))
(complex (random *side*) (random *side*))))
(closed (options-closed *opts*))
(tour
(if closed
(make-closed-tour (list start))
(make-tour (list start)))))
(fresh-line)
(if closed (princ "Closed "))
(princ "Knight's tour")
(if (options-start *opts*)
(princ ":")
(princ " (starting on a random case):"))
(fresh-line)
(print-tour-linear tour)
(princ #\newline)
(princ #\newline)
(print-tour-matrix tour)))

;;; Good to go: invocation!

(prompt)
(main)
```
Output:
```Starting case (leave blank for random)? a8
Require a closed tour (yes or default to no)? y

Closed Knight's tour:
a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> e6 -> d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> e1 -> g2 -> f4 -> d3 -> c5 -> a4 -> b2 -> d1 -> c3 -> b1 -> a3 -> b5 -> a7 -> c6 -> d4 -> f3 -> h4 -> g6 -> h8 -> f7 -> e5 -> g4 -> h2 -> f1 -> d2 -> e4 -> f6 -> g8 -> h6 -> f5 -> e7 -> d5 -> e3 -> c4 -> d6 -> c8 -> b6

1 16 63 22  3 18 55 46
40 23  2 17 58 47  4 19
15 64 41 62 21 54 45 56
24 39 32 59 48 57 20  5
33 14 61 42 53 30 49 44
38 25 36 31 60 43  6  9
13 34 27 52 11  8 29 50
26 37 12 35 28 51 10  7```

## Clojure

Using warnsdorff's rule

```(defn isin? [x li]
(not= [] (filter #(= x %) li)))

(defn options [movements pmoves n]
(let [x (first (last movements)) y (second (last movements))
op (vec (map #(vector (+ x (first %)) (+ y (second %))) pmoves))
vop (filter #(and (>= (first %) 0) (>= (last %) 0)) op)
vop1 (filter #(and (< (first %) n) (< (last %) n)) vop)]
(vec (filter #(not (isin? % movements)) vop1))))

(defn next-move [movements pmoves n]
(let [op (options movements pmoves n)
sp (map #(vector % (count (options (conj movements %) pmoves n))) op)
m (apply min (map last sp))]
(first (rand-nth (filter #(= m (last %)) sp)))))

(defn jumps [n pos]
(let [movements (vector pos)
pmoves [[1 2] [1 -2] [2 1] [2 -1]
[-1 2] [-1 -2] [-2 -1] [-2 1]]]
(loop [mov movements x 1]
(if (= x (* n n))
mov
(let [np (next-move mov pmoves n)]
(recur (conj mov np) (inc x)))))))
```
Output:
```(jumps 5 [0 0])
[[0 0] [1 2] [0 4] [2 3] [4 4] [3 2] [4 0] [2 1] [1 3] [0 1] [2 0] [4 1] [3 3] [1 4] [0 2] [1 0] [3 1] [4 3] [2 4] [0 3] [1 1] [3 0] [4 2] [3 4] [2 2]]

(jumps 8 [0 0])
[[0 0] [2 1] [4 0] [6 1] [7 3] [6 5] [7 7] [5 6] [3 7] [1 6] [0 4] [1 2] [2 0] [0 1] [1 3] [0 5] [1 7] [2 5] [0 6] [2 7] [4 6] [6 7] [7 5] [6 3] [7 1] [5 0] [3 1] [1 0] [0 2] [1 4] [3 5] [4 7] [6 6] [7 4] [6 2] [7 0] [5 1] [7 2] [6 0] [4 1] [5 3] [3 2] [4 4] [5 2] [3 3] [5 4] [4 2] [2 3] [1 1] [3 0] [2 2] [0 3] [2 4] [4 3] [6 4] [4 5] [2 6] [0 7] [1 5] [3 4] [5 5] [7 6] [5 7] [3 6]]

(let [j (jumps 40 [0 0])]        ;; are
(and (distinct? j)             ;; all squares only once? and
(= (count j) (* 40 40)))) ;; 40*40 squares?
true
```

## CoffeeScript

This algorithm finds 100,000 distinct solutions to the 8x8 problem in about 30 seconds. It precomputes knight moves up front, so it turns into a pure graph traversal problem. The program uses iteration and backtracking to find solutions.

```graph_tours = (graph, max_num_solutions) ->
# graph is an array of arrays
# graph[3] = [4, 5] means nodes 4 and 5 are reachable from node 3
#
# Returns an array of tours (up to max_num_solutions in size), where
# each tour is an array of nodes visited in order, and where each
# tour visits every node in the graph exactly once.
#
complete_tours = []
visited = (false for node in graph)
dead_ends = ({} for node in graph)
tour = [0]

valid_neighbors = (i) ->
arr = []
for neighbor in graph[i]
continue if visited[neighbor]
arr.push neighbor
arr

next_square_to_visit = (i) ->
arr = valid_neighbors i
return null if arr.length == 0

# We traverse to our neighbor who has the fewest neighbors itself.
fewest_neighbors = valid_neighbors(arr[0]).length
neighbor = arr[0]
for i in [1...arr.length]
n = valid_neighbors(arr[i]).length
if n < fewest_neighbors
fewest_neighbors = n
neighbor = arr[i]
neighbor

while tour.length > 0
current_square = tour[tour.length - 1]
visited[current_square] = true
next_square = next_square_to_visit current_square
if next_square?
tour.push next_square
if tour.length == graph.length
complete_tours.push (n for n in tour) # clone
break if complete_tours.length == max_num_solutions
# pessimistically call this a dead end
current_square = next_square
else
# we backtrack
doomed_square = tour.pop()
visited[doomed_square] = false
complete_tours

knight_graph = (board_width) ->
# Turn the Knight's Tour into a pure graph-traversal problem
# by precomputing all the legal moves.  Returns an array of arrays,
# where each element in any subarray is the index of a reachable node.
index = (i, j) ->
# index squares from 0 to n*n - 1
board_width * i + j

reachable_squares = (i, j) ->
deltas = [
[ 1,  2]
[ 1, -2]
[ 2,  1]
[ 2, -1]
[-1,  2]
[-1, -2]
[-2,  1]
[-2, -1]
]
neighbors = []
for delta in deltas
[di, dj] = delta
ii = i + di
jj = j + dj
if 0 <= ii < board_width
if 0 <= jj < board_width
neighbors.push index(ii, jj)
neighbors

graph = []
for i in [0...board_width]
for j in [0...board_width]
graph[index(i, j)] = reachable_squares i, j
graph

illustrate_knights_tour = (tour, board_width) ->
pad = (n) ->
return " _" if !n?
return " " + n if n < 10
"#{n}"

console.log "\n------"
moves = {}
for square, i in tour
moves[square] = i + 1
for i in [0...board_width]
s = ''
for j in [0...board_width]
s += "  " + pad moves[i*board_width + j]
console.log s

BOARD_WIDTH = 8
MAX_NUM_SOLUTIONS = 100000

graph = knight_graph BOARD_WIDTH
tours = graph_tours graph, MAX_NUM_SOLUTIONS
console.log "#{tours.length} tours found (showing first and last)"
illustrate_knights_tour tours[0], BOARD_WIDTH
illustrate_knights_tour tours.pop(), BOARD_WIDTH
```

output

```> time coffee knight.coffee
100000 tours found (showing first and last)

------
1   4  57  20  47   6  49  22
34  19   2   5  58  21  46   7
3  56  35  60  37  48  23  50
18  33  38  55  52  59   8  45
39  14  53  36  61  44  51  24
32  17  40  43  54  27  62   9
13  42  15  30  11  64  25  28
16  31  12  41  26  29  10  63

------
1   4  41  20  63   6  61  22
34  19   2   5  42  21  44   7
3  40  35  64  37  62  23  60
18  33  38  47  56  43   8  45
39  14  57  36  49  46  59  24
32  17  48  55  58  27  50   9
13  54  15  30  11  52  25  28
16  31  12  53  26  29  10  51

real	0m29.741s
user	0m25.656s
sys	0m0.253s
```

## D

### Fast Version

Translation of: C++
```import std.stdio, std.algorithm, std.random, std.range,
std.conv, std.typecons, std.typetuple;

int[N][N] knightTour(size_t N=8)(in string start)
in {
assert(start.length >= 2);
} body {
static struct P { int x, y; }

immutable P[8] moves = [P(2,1), P(1,2), P(-1,2), P(-2,1),
P(-2,-1), P(-1,-2), P(1,-2), P(2,-1)];
int[N][N] data;

int[8] sortMoves(in int x, in int y) {
int[2][8] counts;
foreach (immutable i, immutable ref d1; moves) {
int c = 0;
foreach (immutable ref d2; moves) {
immutable p = P(x + d1.x + d2.x, y + d1.y + d2.y);
if (p.x >= 0 && p.x < N && p.y >= 0 && p.y < N &&
data[p.y][p.x] == 0)
c++;
}
counts[i] = [c, i];
}

counts[].randomShuffle; // Shuffle to randomly break ties.
counts[].sort(); // Lexicographic sort.

int[8] result = void;
transversal(counts[], 1).copy(result[]);
return result;
}

immutable p0 = P(start[0] - 'a', N - to!int(start[1 .. \$]));
data[p0.y][p0.x] = 1;

Tuple!(int, int, int, int[8])[N * N] order;
order[0] = tuple(p0.x, p0.y, 0, sortMoves(p0.x, p0.y));

int n = 0;
while (n < (N * N - 1)) {
immutable int x = order[n][0];
immutable int y = order[n][1];
bool ok = false;
foreach (immutable i; order[n][2] .. 8) {
immutable P d = moves[order[n][3][i]];
if (x+d.x < 0 || x+d.x >= N || y+d.y < 0 || y+d.y >= N)
continue;

if (data[y + d.y][x + d.x] == 0) {
order[n][2] = i + 1;
n++;
data[y + d.y][x + d.x] = n + 1;
order[n] = tuple(x+d.x,y+d.y,0,sortMoves(x+d.x,y+d.y));
ok = true;
break;
}
}

if (!ok) { // Failed. Backtrack.
data[y][x] = 0;
n--;
}
}

return data;
}

void main() {
foreach (immutable i, side; TypeTuple!(5, 8, 31, 101)) {
immutable form = "%(%" ~ text(side ^^ 2).length.text ~ "d %)";
foreach (ref row; ["c3", "b5", "a1", "a1"][i].knightTour!side)
writefln(form, row);
writeln();
}
}
```
Output:
```23 16 11  6 21
10  5 22 17 12
15 24  1 20  7
4  9 18 13  2
25 14  3  8 19

63 20  3 24 59 36  5 26
2 23 64 37  4 25 58 35
19 62 21 50 55 60 27  6
22  1 54 61 38 45 34 57
53 18 49 44 51 56  7 28
12 15 52 39 46 31 42 33
17 48 13 10 43 40 29  8
14 11 16 47 30  9 32 41

275 112  19 116 277 604  21 118 823 770  23 120 961 940  25 122 943 926  27 124 917 898  29 126 911 872  31 128 197 870  33
18 115 276 601  20 117 772 767  22 119 958 851  24 121 954 941  26 123 936 925  28 125 912 899  30 127 910 871  32 129 198
111 274 113 278 605 760 603 822 771 824 769 948 957 960 939 944 953 942 927 916 929 918 897 908 913 900 873 196 875  34 869
114  17 600 273 602 775 766 773 768 949 850 959 852 947 952 955 932 937 930 935 924 915 920 905 894 909 882 901 868 199 130
271 110 279 606 759 610 761 776 821 764 825 816 951 956 853 938 945 934 923 928 919 896 893 914 907 904 867 874 195 876  35
16 581 272 599 280 607 774 765 762 779 950 849 826 815 946 933 854 931 844 857 890 921 906 895 886 883 902 881 200 131 194
109 270 281 580 609 758 611 744 777 820 763 780 817 848 827 808 811 846 855 922 843 858 889 892 903 866 885 192 877  36 201
282  15 582 269 598 579 608 757 688 745 778 819 754 783 814 847 828 807 810 845 856 891 842 859 884 887 880 863 202 193 132
267 108 283 578 583 612 689 614 743 756 691 746 781 818 753 784 809 812 829 806 801 840 835 888 865 862 203 878 191 530  37
14 569 268 585 284 597 576 619 690 687 742 755 692 747 782 813 752 785 802 793 830 805 860 841 836 879 864 529 204 133 190
107 266 285 570 577 584 613 686 615 620 695 684 741 732 711 748 739 794 751 786 803 800 839 834 861 528 837 188 531  38 205
286  13 568 265 586 575 596 591 618 685 616 655 696 693 740 733 712 749 738 795 792 831 804 799 838 833 722 527 206 189 134
263 106 287 508 571 590 587 574 621 592 639 694 683 656 731 710 715 734 787 750 737 796 791 832 721 798 207 532 187 474  39
12 417 264 567 288 509 572 595 588 617 654 657 640 697 680 713 730 709 716 735 788 727 720 797 790 723 526 473 208 135 186
105 262 289 416 507 566 589 512 573 622 593 638 653 682 659 698 679 714 729 708 717 736 789 726 719 472 533 184 475  40 209
290  11 418 261 502 415 510 565 594 513 562 641 658 637 652 681 660 699 678 669 728 707 718 675 724 525 704 471 210 185 136
259 104 291 414 419 506 503 514 511 564 623 548 561 642 551 636 651 670 661 700 677 674 725 706 703 534 211 476 183 396  41
10 331 260 493 292 501 420 495 504 515 498 563 624 549 560 643 662 635 650 671 668 701 676 673 524 705 470 395 212 137 182
103 258 293 330 413 494 505 500 455 496 547 516 485 552 625 550 559 644 663 634 649 672 667 702 535 394 477 180 397  42 213
294   9 332 257 492 329 456 421 490 499 458 497 546 517 484 553 626 543 558 645 664 633 648 523 666 469 536 393 220 181 138
255 102 295 328 333 412 491 438 457 454 489 440 459 486 545 518 483 554 627 542 557 646 665 632 537 478 221 398 179 214  43
8 319 256 335 296 345 326 409 422 439 436 453 488 441 460 451 544 519 482 555 628 541 522 647 468 631 392 219 222 139 178
101 254 297 320 327 334 411 346 437 408 423 368 435 452 487 442 461 450 445 520 481 556 629 538 479 466 399 176 215  44 165
298   7 318 253 336 325 344 349 410 347 360 407 424 383 434 427 446 443 462 449 540 521 480 467 630 391 218 223 164 177 140
251 100 303 300 321 316 337 324 343 350 369 382 367 406 425 384 433 428 447 444 463 430 539 390 465 400 175 216 169 166  45
6 299 252 317 304 301 322 315 348 361 342 359 370 381 366 405 426 385 432 429 448 389 464 401 174 217 224 163 150 141 168
99 250 241 302 235 248 307 338 323 314 351 362 341 358 371 380 365 404 377 386 431 402 173 388 225 160 153 170 167  46 143
240   5  98 249 242 305 234 247 308 339 232 313 352 363 230 357 372 379 228 403 376 387 226 159 154 171 162 149 142 151  82
63   2 239  66  97 236 243 306 233 246 309 340 231 312 353 364 229 356 373 378 227 158 375 172 161 148 155 152  83 144  47
4  67  64  61 238  69  96  59 244  71  94  57 310  73  92  55 354  75  90  53 374  77  88  51 156  79  86  49 146  81  84
1  62   3  68  65  60 237  70  95  58 245  72  93  56 311  74  91  54 355  76  89  52 157  78  87  50 147  80  85  48 145```

### Shorter Version

```import std.stdio, std.math, std.algorithm, std.range, std.typecons;

alias Square = Tuple!(int,"x", int,"y");

const(Square)[] knightTour(in Square[] board, in Square[] moves) pure @safe nothrow {
enum findMoves = (in Square sq) pure nothrow @safe =>
cartesianProduct([1, -1, 2, -2], [1, -1, 2, -2])
.filter!(ij => ij[0].abs != ij[1].abs)
.map!(ij => Square(sq.x + ij[0], sq.y + ij[1]))
.filter!(s => board.canFind(s) && !moves.canFind(s));
auto newMoves = findMoves(moves.back);
if (newMoves.empty)
return moves;
//alias warnsdorff = min!(s => findMoves(s).walkLength);
//immutable newSq = newMoves.dropOne.fold!warnsdorff(newMoves.front);
auto pairs = newMoves.map!(s => tuple(findMoves(s).walkLength, s));
immutable newSq = reduce!min(pairs.front, pairs.dropOne)[1];
return board.knightTour(moves ~ newSq);
}

void main(in string[] args) {
enum toSq = (in string xy) => Square(xy[0] - '`', xy[1] - '0');
immutable toAlg = (in Square s) => [dchar(s.x + '`'), dchar(s.y + '0')];
immutable sq = toSq((args.length == 2) ? args[1] : "e5");
const board = iota(1, 9).cartesianProduct(iota(1, 9)).map!Square.array;
writefln("%(%-(%s -> %)\n%)", board.knightTour([sq]).map!toAlg.chunks(8));
}
```
Output:
```e5 -> d7 -> b8 -> a6 -> b4 -> a2 -> c1 -> b3
a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4
h6 -> g8 -> e7 -> c8 -> a7 -> c6 -> a5 -> b7
d8 -> f7 -> h8 -> g6 -> f8 -> h7 -> f6 -> e8
g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4
b6 -> a8 -> c7 -> b5 -> c3 -> d5 -> e3 -> c4
d6 -> e4 -> c5 -> d3 -> e1 -> g2 -> h4 -> f5
d4 -> e2 -> f4 -> e6 -> g5 -> f3 -> g1 -> h3```

## Delphi

Works with: Delphi version 6.0

Brute force method. Takes a long time for most solutions, so some optimization should be used. However, it has nice graphics.

```{ These routines would normally be in a library,
but are presented here for clarity }

function PointAdd(V1,V2: TPoint): TPoint;
{Add V1 and V2}
begin
Result.X:= V1.X+V2.X;
Result.Y:= V1.Y+V2.Y;
end;

const KnightMoves: array [0..7] of TPoint = (
(X: 2; Y:1),(X: 2; Y:-1),
(X:-2; Y:1),(X:-2; Y:-1),
(X:1; Y: 2),(X:-1; Y: 2),
(X:1; Y:-2),(X:-1; Y:-2));

var Board: array [0..7,0..7] of boolean;

var Path: array of TPoint;

var CellSize,BoardSize: integer;

var CurPos: TPoint;

var BestPath: integer;

{-------------------------------------------------------------}

procedure DrawBestPath(Image: TImage);
begin
Image.Canvas.TextOut(BoardSize+5,5, IntToStr(BestPath));
end;

procedure PushPath(P: TPoint);
begin
SetLength(Path,Length(Path)+1);
Path[High(Path)]:=P;
if Length(Path)>BestPath then BestPath:=Length(Path);
end;

function PopPath: TPoint;
begin
if Length(Path)<1 then exit;
Result:=Path[High(Path)];
SetLength(Path,Length(Path)-1);
end;

procedure ClearPath;
begin
SetLength(Path,0);
end;

{-------- Routines to draw chess board and path --------------}

function GetCellCenter(P: TPoint): TPoint;
{Get pixel position of the center of cell}
begin
Result.X:=CellSize div 2 + CellSize * P.X;
Result.Y:=CellSize div 2 + CellSize * P.Y;
end;

procedure DrawPoint(Canvas: TCanvas; P: TPoint);
{Draw a point on the board}
begin
Canvas.Pen.Color:=clYellow;
Canvas.MoveTo(P.X-1,P.Y-1);
Canvas.LineTo(P.X+1,P.Y+1);
Canvas.MoveTo(P.X+1,P.Y-1);
Canvas.LineTo(P.X-1,P.Y+1);
end;

procedure DrawPathLine(Canvas: TCanvas; P1,P2: TPoint);
{Draw the path line}
var PS1,PS2: TPoint;
begin
PS1:=GetCellCenter(P1);
PS2:=GetCellCenter(P2);
Canvas.Pen.Width:=5;
Canvas.Pen.Color:=clRed;
Canvas.MoveTo(PS1.X,PS1.Y);
Canvas.LineTo(PS2.X,PS2.Y);
DrawPoint(Canvas,PS1);
DrawPoint(Canvas,PS2);
end;

procedure DrawPath(Canvas: TCanvas);
{Draw all points on the path}
var I: integer;
begin
for I:=0 to High(Path)-1 do
begin
DrawPathLine(Canvas, Path[I],Path[I+1]);
end;
end;

procedure DrawBoard(Canvas: TCanvas);
{Draw the chess board}
var R,R2: TRect;
var X,Y: integer;
var Color: TColor;
begin
Canvas.Pen.Color:=clBlack;
R:=Rect(0,0,BoardSize,BoardSize);
Canvas.Rectangle(R);
R:=Rect(0,0,CellSize,CellSize);
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
begin
R2:=R;
if ((X+Y) mod 2)=0 then Color:=clWhite
else Color:=clBlack;
Canvas.Brush.Color:=Color;
OffsetRect(R2,X * CellSize, Y * CellSize);
Canvas.Rectangle(R2);
end;
DrawPath(Canvas);
end;

function AllVisited: boolean;
{Test if all squares have been visit by path}
var X,Y: integer;
begin
Result:=False;
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
if not Board[X,Y] then exit;
Result:=True;
end;

procedure ClearBoard;
{Clear all board positions}
var X,Y: integer;
begin
for Y:=0 to High(Board[0]) do
for X:=0 to High(Board) do
Board[X,Y]:=False;
end;

function IsValidMove(Pos,Move: TPoint): boolean;
{Test if potential move is valid}
var NP: TPoint;
begin
Result:=False;
if (NP.X<0) or (NP.X>High(Board)) or
(NP.Y<0) or (NP.Y>High(Board[0])) then exit;
if Board[NP.X,NP.Y] then exit;
Result:=True;
end;

procedure ConfigureScreen(Image: TImage);
{Configure screen size}
begin
if Image.Width<Image.Height then BoardSize:=Image.Width
else BoardSize:=Image.Height;
CellSize:=BoardSize div 8;
end;

procedure SetPosition(Image: TImage; P: TPoint; Value: boolean);
{Set a new position by adding it to path}
{Marking position as used and redrawing board}
begin
if Value then PushPath(P)
else P:=PopPath;
Board[P.X,P.Y]:=Value;
DrawBoard(Image.Canvas);
DrawBestPath(Image);
Image.Repaint;
end;

procedure TryAllMoves(Image: TImage; Pos: TPoint);
{Recursively try all moves}
var I: integer;
var NewPos: TPoint;
begin
SetPosition(Image,Pos,True);
if AllVisited then exit;
for I:=0 to High(KnightMoves) do
begin
if AbortFlag then Exit;
if IsValidMove(Pos,KnightMoves[I]) then
begin
TryAllMoves(Image,NewPos);
end;
end;
SetPosition(Image,Pos,False);
Application.ProcessMessages;
end;

procedure DoKnightsTour(Image: TImage);
{Solve Knights tour by testing all paths}
begin
BestPath:=0;
ConfigureScreen(Image);
ClearPath;
ClearBoard;
DrawBoard(Image.Canvas);
TryAllMoves(Image, Point(0,0));
end;
```
Output:

## EchoLisp

The algorithm uses iterative backtracking and Warnsdorff's heuristic. It can output closed or non-closed tours.

```(require 'plot)
(define *knight-moves*
'((2 . 1)(2 . -1 ) (1 . -2) (-1 . -2  )(-2 . -1) (-2 . 1) (-1 . 2) (1 . 2)))
(define *hit-squares* null)
(define *legal-moves* null)
(define *tries* 0)

(define (square x y n ) (+ y (* x n)))
(define (dim n) (1- (* n n))) ; n^2 - 1

;; check legal knight move from sq
;; return null or (list destination-square)

(define (legal-disp n sq k-move)
(let ((x (+ (quotient sq n) (first k-move)))
(y (+  (modulo sq n)  (rest k-move))))
(if (and (>= x 0) (< x n) (>= y 0) (< y n))
(list (square x y n))  null)))

;; list of legal destination squares from sq
(define (legal-moves  sq  k-moves n )
(if (null? k-moves) null
(append (legal-moves sq (rest k-moves) n) (legal-disp n sq (first k-moves)))))

;; square freedom = number of destination squares not already reached
(define (freedom sq)
(for/sum ((dest (vector-ref *legal-moves* sq)))
(if (vector-ref *hit-squares* dest) 0 1)))

;; The chess adage" A knight on the rim is dim" is false here :
;; choose to move to square with smallest freedom : Warnsdorf's rule
(define (square-sort a b)
(< (freedom a) (freedom b)))

;; knight tour engine
(define (play sq step starter last-one wants-open)
(set! *tries* (1+ *tries*))
(vector-set! *hit-squares* sq step) ;; flag used square
(if (= step last-one) (throw 'HIT last-one)) ;; stop on first path found

(when (or wants-open ;; cut search iff closed path
(and  (< step last-one) (> (freedom starter) 0))) ;; this ensures a closed path

(for ((target (list-sort square-sort (vector-ref *legal-moves* sq))))
(unless (vector-ref *hit-squares* target)
(play target (1+ step)  starter last-one wants-open))))
(vector-set! *hit-squares* sq #f)) ;; unflag used square

(define (show-steps n wants-open)
(string-delimiter "")
(if wants-open
(printf "♘-tour: %d tries."  *tries*)
(printf "♞-closed-tour: %d tries."  *tries*))
(for ((x n))
(writeln)
(for((y n))
(write (string-pad-right (vector-ref *hit-squares*  (square x y n)) 4)))))

(define (k-tour (n  8) (starter 0) (wants-open #t))
(set! *hit-squares* (make-vector (* n n) #f))
;; build vector of legal moves for squares 0..n^2-1
(set! *legal-moves*
(build-vector (* n n) (lambda(sq) (legal-moves sq *knight-moves* n))))
(set! *tries* 0) ; counter
(try
(play starter 0 starter (dim n) wants-open)
(catch (hit mess) (show-steps n wants-open))))
```

Output:
```(k-tour 8 0 #f)
♞-closed-tour: 66 tries.
0   47  14  31  62  27  12  29
15  32  63  54  13  30  57  26
48  1   46  61  56  59  28  11
33  16  55  50  53  44  25  58
2   49  42  45  60  51  10  39
17  34  19  52  43  40  7   24
20  3   36  41  22  5   38  9
35  18  21  4   37  8   23  6

(k-tour 20 57)
♘-tour: 400 tries.
31  34  29  104 209 36  215 300 211 38  213 354 343 40  345 386 383 42  1   388
28  103 32  35  216 299 210 37  214 335 342 39  346 385 382 41  390 387 396 43
33  30  105 208 201 308 301 336 323 212 353 340 355 344 391 384 395 0   389 2
102 27  202 219 298 217 322 309 334 341 356 347 358 351 376 381 378 399 44  397
203 106 207 200 307 228 311 302 337 324 339 352 373 364 379 392 375 394 3   368
26  101 220 229 218 297 304 321 310 333 348 357 350 359 374 377 380 367 398 45
107 204 199 206 227 306 231 312 303 338 325 330 363 372 365 328 393 254 369 4
100 25  122 221 230 233 296 305 320 313 332 349 326 329 360 371 366 251 46  253
121 108 205 198 145 226 237 232 295 286 319 314 331 362 327 316 255 370 5   178
24  99  144 123 222 129 234 279 236 281 294 289 318 315 256 361 250 179 252 47
109 120 111 130 197 146 225 238 285 278 287 272 293 290 317 180 257 162 177 6
98  23  124 143 128 223 276 235 280 239 282 291 288 265 270 249 176 181 48  161
115 110 119 112 131 196 147 224 277 284 273 266 271 292 245 258 163 174 7   58
22  97  114 125 142 127 140 275 194 267 240 283 264 269 248 175 182 59  160 49
87  116 95  118 113 132 195 148 187 274 263 268 191 244 259 246 173 164 57  8
96  21  88  133 126 141 150 139 262 193 190 241 260 247 172 183 60  159 50  65
77  86  117 94  89  138 135 188 149 186 261 192 171 184 243 156 165 64  9   56
20  81  78  85  134 93  90  151 136 189 170 185 242 155 166 61  158 53  66  51
79  76  83  18  91  74  137 16  169 72  153 14  167 70  157 12  63  68  55  10
82  19  80  75  84  17  92  73  152 15  168 71  154 13  62  69  54  11  52  67
```
Plotting

64 shades of gray. We plot the move sequence in shades of gray, from black to white. The starting square is red. The ending square is green. One can observe that the squares near the border are played first (dark squares).

```(define (step-color x y n last-one)
(letrec ((sq (square (floor x) (floor y) n))
(step (vector-ref *hit-squares* sq) n n))
(cond ((= 0 step) (rgb 1 0 0)) ;; red starter
((= last-one step) (rgb 0 1 0)) ;; green end
(else (gray (// step n n))))))

(define  ( k-plot n)
(plot-rgb (lambda (x y) (step-color x y n (dim n))) (- n epsilon) (- n epsilon)))
```

Closed path on a 12x12 board: [1]

Open path on a 24x24 board: [2]

## Elixir

Translation of: Ruby
```defmodule Board do
import Integer, only: [is_odd: 1]

defmodule Cell do
end

defp initialize(rows, cols) do
board = for i <- 1..rows, j <- 1..cols, into: %{}, do: {{i,j}, true}
for i <- 1..rows, j <- 1..cols, into: %{} do
adj = for [di,dj] <- @adjacent, board[{i+di, j+dj}], do: {i+di, j+dj}
end
end

defp solve(board, ij, num, goal) do
board = Map.update!(board, ij, fn cell -> %{cell | value: num} end)
if num == goal do
throw({:ok, board})
else
wdof(board, ij)
|> Enum.each(fn k -> solve(board, k, num+1, goal) end)
end
end

defp wdof(board, ij) do               # Warnsdorf's rule
|> Enum.filter(fn k -> board[k].value == 0 end)
|> Enum.sort_by(fn k ->
Enum.count(board[k].adj, fn x -> board[x].value == 0 end)
end)
end

defp to_string(board, rows, cols) do
width = to_string(rows * cols) |> String.length
format = String.duplicate("~#{width}w ", cols)
Enum.map_join(1..rows, "\n", fn i ->
:io_lib.fwrite format, (for j <- 1..cols, do: board[{i,j}].value)
end)
end

def knight_tour(rows, cols, sx, sy) do
IO.puts "\nBoard (#{rows} x #{cols}), Start: [#{sx}, #{sy}]"
if is_odd(rows*cols) and is_odd(sx+sy) do
IO.puts "No solution"
else
try do
initialize(rows, cols)
|> solve({sx,sy}, 1, rows*cols)
IO.puts "No solution"
catch
{:ok, board} -> IO.puts to_string(board, rows, cols)
end
end
end
end

Board.knight_tour(8,8,4,2)
Board.knight_tour(5,5,3,3)
Board.knight_tour(4,9,1,1)
Board.knight_tour(5,5,1,2)
Board.knight_tour(12,12,2,2)
```
Output:
```Board (8 x 8), Start: [4, 2]
23 20  3 32 25 10  5  8
2 35 24 21  4  7 26 11
19 22 33 36 31 28  9  6
34  1 50 29 48 37 12 27
51 18 53 44 61 30 47 38
54 43 56 49 58 45 62 13
17 52 41 60 15 64 39 46
42 55 16 57 40 59 14 63

Board (5 x 5), Start: [3, 3]
19  8  3 14 25
2 13 18  9  4
7 20  1 24 15
12 17 22  5 10
21  6 11 16 23

Board (4 x 9), Start: [1, 1]
1 34  3 28 13 24  9 20 17
4 29  6 33  8 27 18 23 10
35  2 31 14 25 12 21 16 19
30  5 36  7 32 15 26 11 22

Board (5 x 5), Start: [1, 2]
No solution

Board (12 x 12), Start: [2, 2]
87  24  59   2  89  26  61   4  39   8  31   6
58   1  88  25  60   3  92  27  30   5  38   9
23  86  83  90 103  98  29  62  93  40   7  32
82  57 102  99  84  91 104  97  28  37  10  41
101  22  85 114 105 116 111  94  63  96  33  36
56  81 100 123 128 113 106 117 110  35  42  11
21 122 141  80 115 124 127 112  95  64 109  34
144  55  78 121 142 129 118 107 126 133  12  43
51  20 143 140  79 120 125 138  69 108  65 134
54  73  52  77 130 139  70 119 132 137  44  13
19  50  75  72  17  48 131  68  15  46 135  66
74  53  18  49  76  71  16  47 136  67  14  45
```

## Elm

```module Main exposing (main)

import Browser exposing (element)
import Html as H
import Html.Attributes as HA
import List exposing (filter, head, length, map, map2, member, tail)
import List.Extra exposing (andThen, minimumBy)
import String exposing (join)
import Svg exposing (g, line, rect, svg)
import Svg.Attributes exposing (fill, height, style, version, viewBox, width, x, x1, x2, y, y1, y2)
import Svg.Events exposing (onClick)
import Time exposing (every)
import Tuple

type alias Cell =
( Int, Int )

type alias BoardSize =
( Int, Int )

type alias Model =
{ path : List Cell
, board : List Cell
, pause_ms : Float
, size : BoardSize
}

type Msg
= Tick Time.Posix
| SetStart Cell
| SetSize BoardSize
| SetPause Float

boardsize_width: BoardSize -> Int
boardsize_width bs =
Tuple.second bs

boardsize_height: BoardSize -> Int
boardsize_height bs =
Tuple.first bs

boardsize_dec: Int -> Int
boardsize_dec n =
let
minimum_size = 3
in
if n <= minimum_size then
minimum_size
else
n - 1
boardsize_inc: Int -> Int
boardsize_inc n =
let
maximum_size = 40
in
if n >= maximum_size then
maximum_size
else
n + 1

pause_inc: Float -> Float
pause_inc n =
n + 10

-- decreasing pause time (ms) increases speed
pause_dec: Float -> Float
pause_dec n =
let
minimum_pause = 0
in
if n <= minimum_pause then
minimum_pause
else
n - 10

board_init : BoardSize -> List Cell
board_init board_size =
List.range 0 (boardsize_height board_size - 1)
|> andThen
(\r ->
List.range 0 (boardsize_width board_size - 1)
|> andThen
(\c ->
[ ( r, c ) ]
)
)

nextMoves : Model -> Cell -> List Cell
nextMoves model ( stRow, stCol ) =
let
c =
[ 1, 2, -1, -2 ]

km =
c
|> andThen
(\cRow ->
c
|> andThen
(\cCol ->
if abs cRow == abs cCol then
[]

else
[ ( cRow, cCol ) ]
)
)

jumps =
List.map (\( kmRow, kmCol ) -> ( kmRow + stRow, kmCol + stCol )) km
in
List.filter (\j -> List.member j model.board && not (List.member j model.path)) jumps

bestMove : Model -> Maybe Cell
bestMove model =
case List.head model.path of
Just mph ->
minimumBy (List.length << nextMoves model) (nextMoves model mph)
_ ->
Nothing

-- Initialize the application - https://guide.elm-lang.org/effects/
init : () -> ( Model, Cmd Msg )
init _ =
let
-- Initial board height and width
initial_size =
8

-- Initial chess board
initial_board =
board_init (initial_size, initial_size)

initial_path =
[]
initial_pause =
10
in
( Model initial_path initial_board initial_pause (initial_size, initial_size), Cmd.none )

-- View the model - https://guide.elm-lang.org/effects/
view : Model -> H.Html Msg
view model =
let
showChecker row col =
rect
[ x <| String.fromInt col
, y <| String.fromInt row
, width "1"
, height "1"
, fill <|
if modBy 2 (row + col) == 0 then
"blue"

else
"grey"
, onClick <| SetStart ( row, col )
]
[]

showMove ( row0, col0 ) ( row1, col1 ) =
line
[ x1 <| String.fromFloat (toFloat col0 + 0.5)
, y1 <| String.fromFloat (toFloat row0 + 0.5)
, x2 <| String.fromFloat (toFloat col1 + 0.5)
, y2 <| String.fromFloat (toFloat row1 + 0.5)
, style "stroke:yellow;stroke-width:0.05"
]
[]

render mdl =
let
checkers =
mdl.board
|> andThen
(\( r, c ) ->
[ showChecker r c ]
)

moves =
case List.tail mdl.path of
Nothing ->
[]

Just tl ->
List.map2 showMove mdl.path tl
in
checkers ++ moves

unvisited =
length model.board - length model.path

center =
[ HA.style "text-align" "center" ]

table =
[ HA.style "text-align" "center", HA.style "display" "table", HA.style "width" "auto", HA.style "margin" "auto" ]
table_row =
[ HA.style "display" "table-row", HA.style "width" "auto" ]

table_cell =
[ HA.style "display" "table-cell", HA.style "width" "auto", HA.style "padding" "1px 3px" ]
rows =
boardsize_height model.size

cols =
boardsize_width model.size
in
H.div
[]
[ H.h1 center [ H.text "Knight's Tour" ]
-- controls
, H.div
table
[ H.div -- labels
table_row
[ H.div
table_cell
[ H.text "Rows"]
, H.div
table_cell
[ H.text "Columns"]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text "Pause (ms)"]
]
, H.div
table_row
[ H.div -- Increase
table_cell
[ H.button [onClick <| SetSize ( boardsize_inc rows, cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_inc cols )] [ H.text "▲"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_inc model.pause_ms )] [ H.text "▲"] ]
]
, H.div
table_row
[ H.div -- Value
table_cell
[ H.text <| String.fromInt rows ]
, H.div
table_cell
[ H.text <| String.fromInt cols]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.text <| String.fromFloat model.pause_ms]
]
, H.div
table_row
[ H.div -- Decrease
table_cell
[ H.button [onClick <| SetSize ( boardsize_dec rows, cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.button [onClick <| SetSize ( rows, boardsize_dec cols )] [ H.text "▼"] ]
, H.div
table_cell
[ H.text ""]
, H.div
table_cell
[ H.button [onClick <| SetPause ( pause_dec model.pause_ms )] [ H.text "▼"] ]
]
]
, H.h2 center [ H.text "(pick a square)" ]
, H.div -- chess board
center
[ svg
[ version "1.1"
, width (String.fromInt (25 * cols))
, height (String.fromInt (25 * rows))
, viewBox
(join " "
[ String.fromInt 0
, String.fromInt 0
, String.fromInt cols
, String.fromInt rows
]
)
]
[ g [] <| render model ]
]
, H.h3 center [ H.text <| "Unvisited count : " ++ String.fromInt unvisited ]
]

-- Update the model - https://guide.elm-lang.org/effects/
update : Msg -> Model -> ( Model, Cmd Msg )
update msg model =
let
mo =
case msg of
SetPause pause ->
{ model | pause_ms = pause }

SetSize board_size ->
{ model | board = board_init board_size, path = [], size = board_size }

SetStart start ->
{ model | path = [ start ] }

Tick _ ->
case model.path of
[] ->
model

_ ->
case bestMove model of
Nothing ->
model

Just best ->
{ model | path = best :: model.path }
in
( mo, Cmd.none )

-- Subscribe to https://guide.elm-lang.org/effects/
subscriptions : Model -> Sub Msg
subscriptions model =
Time.every model.pause_ms Tick

-- Application entry point
main: Program () Model Msg
main =
element -- https://package.elm-lang.org/packages/elm/browser/latest/Browser#element
{ init = init
, view = view
, update = update
, subscriptions = subscriptions
}
```

Link to live demo: https://dmcbane.github.io/knights-tour/

## Erlang

Again I use backtracking. It seemed easier this time.

```-module( knights_tour ).

-export( [display/1, solve/1, task/0] ).

display( Moves ) ->
%% The knigh walks the moves {Position, Step_nr} order.
%% Top left corner is {\$a, 8}, Bottom right is {\$h, 1}.
io:fwrite( "Moves:" ),
lists:foldl( fun display_moves/2, erlang:length(Moves), lists:keysort(2, Moves) ),
io:nl(),
[display_row(Y, Moves) || Y <- lists:seq(8, 1, -1)].

solve( First_square ) ->
try
bt_loop( 1, next_moves(First_square), [{First_square, 1}] )

catch
_:{ok, Moves} -> Moves

end.

io:fwrite( "Starting {a, 1}~n" ),
Moves = solve( {\$a, 1} ),
display( Moves ).

bt( N, Move, Moves ) -> bt_reject( is_not_allowed_knight_move(Move, Moves), N, Move, [{Move, N} | Moves] ).

bt_accept( true, _N, _Move, Moves ) -> erlang:throw( {ok, Moves} );
bt_accept( false, N, Move, Moves ) -> bt_loop( N, next_moves(Move), Moves ).

bt_loop( N, New_moves, Moves ) -> [bt( N+1, X, Moves ) || X <- New_moves].

bt_reject( true, _N, _Move, _Moves ) -> backtrack;
bt_reject( false, N, Move, Moves ) -> bt_accept( is_all_knights(Moves), N, Move, Moves ).

display_moves( {{X, Y}, 1}, Max ) ->
io:fwrite(" ~p. N~c~p", [1, X, Y]),
Max;
display_moves( {{X, Y}, Max}, Max ) ->
io:fwrite(" N~c~p~n", [X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) when Step_nr rem 8 =:= 0 ->
io:fwrite(" N~c~p~n~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max;
display_moves( {{X, Y}, Step_nr}, Max ) ->
io:fwrite(" N~c~p ~p. N~c~p", [X, Y, Step_nr, X, Y]),
Max.

display_row( Row, Moves ) ->
[io:fwrite(" ~2b", [proplists:get_value({X, Row}, Moves)]) || X <- [\$a, \$b, \$c, \$d, \$e, \$f, \$g, \$h]],
io:nl().

is_all_knights( Moves ) when erlang:length(Moves) =:= 64 -> true;
is_all_knights( _Moves ) -> false.

is_asymetric( Start_column, Start_row, Stop_column, Stop_row ) ->
erlang:abs( Start_column - Stop_column ) =/= erlang:abs( Start_row - Stop_row ).

is_not_allowed_knight_move( Move, Moves ) ->
no_such_move =/= proplists:get_value( Move, Moves, no_such_move ).

next_moves( {Column, Row} ) ->
[{X, Y} || X <- next_moves_column(Column), Y <- next_moves_row(Row), is_asymetric(Column, Row, X, Y)].

next_moves_column( \$a ) -> [\$b, \$c];
next_moves_column( \$b ) -> [\$a, \$c, \$d];
next_moves_column( \$g ) -> [\$e, \$f, \$h];
next_moves_column( \$h ) -> [\$g, \$f];
next_moves_column( C ) -> [C - 2, C - 1, C + 1, C + 2].

next_moves_row( 1 ) -> [2, 3];
next_moves_row( 2 ) -> [1, 3, 4];
next_moves_row( 7 ) -> [5, 6, 8];
next_moves_row( 8 ) -> [6, 7];
next_moves_row( N ) -> [N - 2, N - 1, N + 1, N + 2].
```
Output:
```17> knights_tour:task().
Starting {a, 1}
Moves: 1. Na1 Nb3 2. Nb3 Na5 3. Na5 Nb7 4. Nb7 Nc5 5. Nc5 Na4 6. Na4 Nb2 7. Nb2 Nc4
8. Nc4 Na3 9. Na3 Nb1 10. Nb1 Nc3 11. Nc3 Na2 12. Na2 Nb4 13. Nb4 Na6 14. Na6 Nb8 15. Nb8 Nc6
16. Nc6 Na7 17. Na7 Nb5 18. Nb5 Nc7 19. Nc7 Na8 20. Na8 Nb6 21. Nb6 Nc8 22. Nc8 Nd6 23. Nd6 Ne4
24. Ne4 Nd2 25. Nd2 Nf1 26. Nf1 Ne3 27. Ne3 Nc2 28. Nc2 Nd4 29. Nd4 Ne2 30. Ne2 Nc1 31. Nc1 Nd3
32. Nd3 Ne1 33. Ne1 Ng2 34. Ng2 Nf4 35. Nf4 Nd5 36. Nd5 Ne7 37. Ne7 Ng8 38. Ng8 Nh6 39. Nh6 Nf5
40. Nf5 Nh4 41. Nh4 Ng6 42. Ng6 Nh8 43. Nh8 Nf7 44. Nf7 Nd8 45. Nd8 Ne6 46. Ne6 Nf8 47. Nf8 Nd7
48. Nd7 Ne5 49. Ne5 Ng4 50. Ng4 Nh2 51. Nh2 Nf3 52. Nf3 Ng1 53. Ng1 Nh3 54. Nh3 Ng5 55. Ng5 Nh7
56. Nh7 Nf6 57. Nf6 Ne8 58. Ne8 Ng7 59. Ng7 Nh5 60. Nh5 Ng3 61. Ng3 Nh1 62. Nh1 Nf2 63. Nf2 Nd1

20 15 22 45 58 47 38 43
17  4 19 48 37 44 59 56
14 21 16 23 46 57 42 39
3 18  5 36 49 40 55 60
6 13  8 29 24 35 50 41
9  2 11 32 27 52 61 54
12  7 28 25 30 63 34 51
1 10 31 64 33 26 53 62
```

## ERRE

Taken from ERRE distribution disk. Comments are in Italian.

```! **********************************************************************
! *                                                                    *
! *     IL GIRO DEL CAVALLO - come collocare un cavallo su di una      *
! *                           scacchiera n*n passando una sola volta   *
! *                           per ogni casella.                        *
! *                                                                    *
! **********************************************************************
! ----------------------------------------------------------------------
!                   Inizializzazione dei parametri
! ----------------------------------------------------------------------

PROGRAM KNIGHT

!\$INTEGER
!\$KEY

DIM H[25,25],A[8],B[8],P0[8],P1[8]

!\$INCLUDE="PC.LIB"

PROCEDURE INIT_SCACCHIERA
! **********************************************************************
! *         Routine di inizializzazione scacchiera                     *
! **********************************************************************
FOR I1=1 TO 8 DO
U=X+A[I1]  V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
H[U,V]=H[U,V]-1
END IF
END FOR
END PROCEDURE

PROCEDURE MOSTRA_SCACCHIERA
! *********************************************************************
! *         Routine di visualizzazione della scacchiera               *
! *********************************************************************
LOCATE(5,1)  COLOR(0,7) PRINT(" Mossa num.";NMOS) COLOR(7,0)
L2=N
FOR I2=1 TO N DO
PRINT
FOR L1=1 TO N DO
IF H[L1,L2]>0 THEN COLOR(15,0) END IF
WRITE("####";H[L1,L2];)
COLOR(7,0)
END FOR
L2=L2-1
END FOR
END PROCEDURE

PROCEDURE AGGIORNA_SCACCHIERA
! *********************************************************************
! *        Routine di Aggiornamento Scacchiera                        *
! *********************************************************************
B=1
FOR I1=1 TO 8 DO
U=X+A[I1] V=Y+B[I1]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 THEN
H[U,V]=H[U,V]+1 B=0
END IF
END IF
END FOR
IF B=1 THEN Q1=0 END IF
END PROCEDURE

PROCEDURE MOSSA_MAX_PESO
! *********************************************************************
! *         Cerca la prossima mossa con il massimo peso               *
! *********************************************************************
M1=0  RO=1
FOR W=1 TO 8 DO
U=Z1+A[W] V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]<=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1  T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE

PROCEDURE MOSSA_MIN_PESO
! *********************************************************************
! *          Cerca la prossima mossa con il minimo peso               *
! *********************************************************************
M1=-9 RO=1
FOR W=1 TO 8 DO
U=Z1+A[W]  V=Z2+B[W]
IF (U>0 AND U<=N) AND (V>0 AND V<=N) THEN
IF H[U,V]<=0 AND H[U,V]>=M1 THEN
IF H[U,V]=M1 THEN
RO=RO+1 P0[RO]=W
ELSE
M1=H[U,V] Q1=1  T1=U T2=V RO=1 P0[1]=W
END IF
END IF
END IF
END FOR
END PROCEDURE

BEGIN
A[1]=1     A[2]=2   A[3]=2   A[4]=1
A[5]=-1    A[6]=-2  A[7]=-2  A[8]=-1
B[1]=2     B[2]=1   B[3]=-1  B[4]=-2
B[5]=-2    B[6]=-1  B[7]=1   B[8]=2

CLS
PRINT("            ***    LA GALOPPATA DEL CAVALIERE    ***")
PRINT
PRINT("Inserire la dimensione della scacchiera (max. 25)";)
INPUT(N)
PRINT("Inserire la caselle di partenza (x,y) ";)
INPUT(X1,Y1)
NMOS=1  A1=1  N1=N*N  ESCAPE=FALSE
! ----------------------------------------------------------------------
!                  Set della scacchiera
! ----------------------------------------------------------------------
WHILE NOT ESCAPE DO
FOR I=1 TO N DO
FOR J=1 TO N DO
H[I,J]=0
END FOR
END FOR
FOR I=1 TO N DO
FOR J=1 TO N DO
X=I  Y=J
INIT_SCACCHIERA
END FOR
END FOR

! ----------------------------------------------------------------------
!                       Effettua la prima mossa
! ----------------------------------------------------------------------
X=X1  Y=Y1  H[X,Y]=1   L=2
AGGIORNA_SCACCHIERA
Q1=1  Q2=1
! -----------------------------------------------------------------------
!                        Trova la prossima mossa
! -----------------------------------------------------------------------
WHILE Q1<>0 AND Q2<>0 DO
Q1=0 Z1=X Z2=Y
MOSSA_MIN_PESO
IF RO<=1 THEN
C1=T1    C2=T2
ELSE
! ------------------------------------------------------------------------
!                         Esamina tutti i vincoli
! ------------------------------------------------------------------------
FOR K=1 TO RO DO
P1[K]=P0[K]
END FOR
R1=RO
IF A1=1 THEN M2=-9 ELSE M2=0 END IF
FOR K=1 TO R1 DO
F1=P1[K]   Z1=X+A[F1]   Z2=Y+B[F1]
IF A1=1 THEN
MOSSA_MAX_PESO
IF M1<=M2 THEN
!\$NULL
ELSE
M2=M1 C1=Z1 C2=Z2
END IF
ELSE
MOSSA_MIN_PESO
IF M1>=M2 THEN
!\$NULL
ELSE
M2=M1  C1=Z1  C2=Z2
END IF
END IF
END FOR
! ------------------------------------------------------------------------
!          Prossima mossa trovata:aggiorna la scacchiera
! ------------------------------------------------------------------------
END IF
IF Q1<>0 THEN
X=C1  Y=C2 H[X,Y]=L
AGGIORNA_SCACCHIERA
IF L=N1 THEN Q2=0 END IF
END IF
L=L+1
MOSTRA_SCACCHIERA
NMOS=NMOS+1
END WHILE
! ------------------------------------------------------------------------
!           La ricerca è terminata: visualizza i risultati
! ------------------------------------------------------------------------
PRINT PRINT
IF Q2<>1 THEN
PRINT("*** Trovata la soluzione! ***")
MOSTRA_SCACCHIERA
ESCAPE=TRUE
ELSE
IF A1=0 THEN
PRINT("Nessuna soluzione.")
ESCAPE=TRUE
ELSE
BEEP
A1=0
END IF
END IF
END WHILE
REPEAT
GET(A\$)
UNTIL A\$<>""
END PROGRAM```
Output:
```            ***    LA GALOPPATA DEL CAVALIERE    ***

Inserire la dimensione della scacchiera (max. 25)? 8
Inserire la caselle di partenza (x,y) ? 1,1
Mossa num. 64

64   7  54  41  60   9  48  39
53  42  61   8  55  40  35  10
6  63  44  59  34  49  38  47
43  52  21  62  45  56  11  36
20   5  58  33  50  37  46  25
31   2  51  22  57  26  15  12
4  19  32  29  14  17  24  27
1  30   3  18  23  28  13  16

*** Trovata la soluzione! ***
```

## FreeBASIC

```Dim Shared As Integer tamano, xc, yc, nm
Dim As Integer f, qm, nmov, n = 0
Dim As String posini

Cls : Color 11
Input "Tamaño tablero:  ", tamano
Input "Posicion inicial: ", posini

Dim As Integer x = Asc(Mid(posini,1,1))-96
Dim As Integer y = Val(Mid(posini,2,1))
Dim Shared As Integer tablero(tamano,tamano), dx(8), dy(8)
For f = 1 To 8 : Read dx(f), dy(f) : Next f
Data 2,1,1,2,-1,2,-2,1,-2,-1,-1,-2,1,-2,2,-1

Sub FindMoves()
Dim As Integer i, xt, yt
If xc < 1 Or yc < 1 Or xc > tamano Or yc > tamano Then nm = 1000: Return
If tablero(xc,yc) Then nm = 2000: Return
nm = 0
For i = 1 To 8
xt = xc+dx(i)
yt = yc+dy(i)
If xt < 1 Or yt < 1 Or xt > tamano Or yt > tamano Then 'Salta este movimiento
Elseif tablero(xt,yt) Then 'Salta este movimiento
Else
nm += 1
End If
Next i
End Sub

Color 4, 7 'Pinta tablero
For f = 1 To tamano
Locate 15-tamano, 3*f: Print "  "; Chr(96+f); " ";
Locate 17-f, 3*(tamano+1)+1: Print Using "##"; f;
Next f

Color 15, 0
Do
n += 1
tablero(x,y) = n
Locate 17-y, 3*x: Print Using "###"; n;
If n = tamano*tamano Then Exit Do
nmov = 100
For f = 1 To 8
xc = x+dx(f)
yc = y+dy(f)
FindMoves()
If nm < nmov Then nmov = nm: qm = f
Next f
x = x+dx(qm)
y = y+dy(qm)
Sleep 1
Loop
Color 14 : Locate Csrlin+tamano, 1
Print " Pulsa cualquier tecla para finalizar..."
Sleep
End```
Output:
```Tamaño tablero:  8
Posicion inicial: c3

a  b  c  d  e  f  g  h

24 11 22 19 26  9 38 47  8
21 18 25 10 39 48 27  8  7
12 23 20 53 28 37 46 49  6
17 52 29 40 59 50  7 36  5
30 13 58 51 54 41 62 45  4
57 16  1 42 63 60 35  6  3
2 31 14 55  4 33 44 61  2
15 56  3 32 43 64  5 34  1

Pulsa cualquier tecla para finalizar...
```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website.

In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.

## Fortran

### FORTRAN 77

Translation of: ATS
Works with: gfortran version 11.2.1
Works with: f2c
```C-----------------------------------------------------------------------
C
C     Find Knight’s Tours.
C
C     Using Warnsdorff’s heuristic, find multiple solutions.
C     Optionally accept only closed tours.
C
C     This program is migrated from my implementation for ATS/Postiats.
C     Arrays with dimension 1:64 take the place of stack frames.
C
C     Compile with, for instance:
C
C         gfortran -O2 -g -std=legacy -o knights_tour knights_tour.f
C
C     or
C
C         f2c knights_tour.f
C         cc -O -o knights_tour knights_tour.c -lf2c
C
C     Usage examples:
C
C         One tour starting at a1, either open or closed:
C
C            echo "a1 1 F" | ./knights_tour
C
C         No more than 2000 closed tours starting at c5:
C
C            echo "c5 2000 T" | ./knights_tour
C
C-----------------------------------------------------------------------

program ktour
implicit none

character*2 alg
integer i, j
integer mxtour
logical closed

read (*,*) alg, mxtour, closed
call alg2ij (alg, i, j)
call explor (i, j, mxtour, closed)

end

C-----------------------------------------------------------------------

subroutine explor (istart, jstart, mxtour, closed)
implicit none

C     Explore the space of 'Warnsdorffian' knight’s paths, looking for
C     and printing complete tours.

integer istart, jstart    ! The starting position.
integer mxtour            ! The maximum number of tours to print.
logical closed            ! Closed tours only?

integer board(1:8,1:8)
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer n
integer itours
logical goodmv
logical isclos

itours = 0
call initbd (board)
n = 1
nmove(1) = 8
imove(8, 1) = istart
jmove(8, 1) = jstart

1000 if (itours .lt. mxtour .and. n .ne. 0) then

if (nmove(n) .eq. 9) then
n = n - 1
if (n .ne. 0) then
call unmove (board, imove, jmove, nmove, n)
nmove(n) = nmove(n) + 1
end if
else if (goodmv (imove, nmove, n)) then
call mkmove (board, imove, jmove, nmove, n)
if (n .eq. 64) then
if (.not. closed) then
itours = itours + 1
call prnt (board, itours)
else if (isclos (board)) then
itours = itours + 1
call prnt (board, itours)
end if
call unmove (board, imove, jmove, nmove, n)
nmove(n) = 9
else if (n .eq. 63) then
call possib (board, n, imove, jmove, nmove)
n = n + 1
nmove(n) = 1
else
call nxtmov (board, n, imove, jmove, nmove)
n = n + 1
nmove(n) = 1
end if
else
nmove(n) = nmove(n) + 1
end if

goto 1000
end if

end

C-----------------------------------------------------------------------

subroutine initbd (board)
implicit none

C     Initialize a chessboard with empty squares.

integer board(1:8,1:8)

integer i, j

do 1010 j = 1, 8
do 1000 i = 1, 8
board(i, j) = -1
1000    continue
1010 continue

end

C-----------------------------------------------------------------------

subroutine mkmove (board, imove, jmove, nmove, n)
implicit none

C     Fill a square with a move number.

integer board(1:8, 1:8)
integer imove(1:8, 1:64)
integer jmove(1:8, 1:64)
integer nmove(1:64)
integer n

board(imove(nmove(n), n), jmove(nmove(n), n)) = n

end

C-----------------------------------------------------------------------

subroutine unmove (board, imove, jmove, nmove, n)
implicit none

C     Unmake a mkmove.

integer board(1:8, 1:8)
integer imove(1:8, 1:64)
integer jmove(1:8, 1:64)
integer nmove(1:64)
integer n

board(imove(nmove(n), n), jmove(nmove(n), n)) = -1

end

C-----------------------------------------------------------------------

function goodmv (imove, nmove, n)
implicit none

logical goodmv
integer imove(1:8, 1:64)
integer nmove(1:64)
integer n

goodmv = (imove(nmove(n), n) .ne. -1)

end

C-----------------------------------------------------------------------

subroutine prnt (board, itours)
implicit none

C     Print a knight's tour.

integer board(1:8,1:8)
integer itours

10000 format (1X)

C     The following plethora of format statements seemed a simple way to
C     get this working with f2c. (For gfortran, the 'I0' format
C     sufficed.)
10010 format (1X, "Tour number ", I1)
10020 format (1X, "Tour number ", I2)
10030 format (1X, "Tour number ", I3)
10040 format (1X, "Tour number ", I4)
10050 format (1X, "Tour number ", I5)
10060 format (1X, "Tour number ", I6)
10070 format (1X, "Tour number ", I20)

if (itours .lt. 10) then
write (*, 10010) itours
else if (itours .lt. 100) then
write (*, 10020) itours
else if (itours .lt. 1000) then
write (*, 10030) itours
else if (itours .lt. 10000) then
write (*, 10040) itours
else if (itours .lt. 100000) then
write (*, 10050) itours
else if (itours .lt. 1000000) then
write (*, 10060) itours
else
write (*, 10070) itours
end if
call prntmv (board)
call prntbd (board)
write (*, 10000)

end

C-----------------------------------------------------------------------

subroutine prntbd (board)
implicit none

C     Print a chessboard with the move number in each square.

integer board(1:8,1:8)

integer i, j

10000 format (1X, "    ", 8("+----"), "+")
10010 format (1X, I2, " ", 8(" | ", I2), " | ")
10020 format (1X, "   ", 8("    ", A1))

do 1000 i = 8, 1, -1
write (*, 10000)
write (*, 10010) i, (board(i, j), j = 1, 8)
1000 continue
write (*, 10000)
write (*, 10020) 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'

end

C-----------------------------------------------------------------------

subroutine prntmv (board)
implicit none

C     Print the moves of a knight's path, in algebraic notation.

integer board(1:8,1:8)

integer ipos(1:64)
integer jpos(1:64)
integer numpos
character*2 alg(1:64)
integer columns(1:8)
integer k
integer m

character*72 lines(1:8)

10000 format (1X, A)

call bd2pos (board, ipos, jpos, numpos)

C     Convert the positions to algebraic notation.
do 1000 k = 1, numpos
call ij2alg (ipos(k), jpos(k), alg(k))
1000 continue

C     Fill lines with algebraic notations.
do 1020 m = 1, 8
columns(m) = 1
1020 continue
m = 1
do 1100 k = 1, numpos
lines(m)(columns(m) : columns(m) + 1) = alg(k)(1:2)
columns(m) = columns(m) + 2
if (k .ne. numpos) then
lines(m)(columns(m) : columns(m) + 3) = " -> "
columns(m) = columns(m) + 4
else if (numpos .eq. 64 .and.
\$           ((abs (ipos(numpos) - ipos(1)) .eq. 2
\$           .and. abs (jpos(numpos) - jpos(1)) .eq. 1) .or.
\$           ((abs (ipos(numpos) - ipos(1)) .eq. 1
\$           .and. abs (jpos(numpos) - jpos(1)) .eq. 2)))) then
lines(m)(columns(m) : columns(m) + 8) = " -> cycle"
columns(m) = columns(m) + 9
endif
if (mod (k, 8) .eq. 0) m = m + 1
1100 continue

C     Print the lines that have stuff in them.
do 1200 m = 1, 8
if (columns(m) .ne. 1) then
write (*, 10000) lines(m)(1 : columns(m) - 1)
end if
1200 continue

end

C-----------------------------------------------------------------------

function isclos (board)
implicit none

C     Is a board a closed tour?

logical isclos
integer board(1:8,1:8)
integer ipos(1:64)        ! The i-positions in order.
integer jpos(1:64)        ! The j-positions in order.
integer numpos            ! The number of positions so far.

call bd2pos (board, ipos, jpos, numpos)

isclos = (numpos .eq. 64 .and.
\$     ((abs (ipos(numpos) - ipos(1)) .eq. 2
\$     .and. abs (jpos(numpos) - jpos(1)) .eq. 1) .or.
\$     ((abs (ipos(numpos) - ipos(1)) .eq. 1
\$     .and. abs (jpos(numpos) - jpos(1)) .eq. 2))))

end

C-----------------------------------------------------------------------

subroutine bd2pos (board, ipos, jpos, numpos)
implicit none

C     Convert from a board to a list of board positions.

integer board(1:8,1:8)
integer ipos(1:64)        ! The i-positions in order.
integer jpos(1:64)        ! The j-positions in order.
integer numpos            ! The number of positions so far.

integer i, j

numpos = 0
do 1010 i = 1, 8
do 1000 j = 1, 8
if (board(i, j) .ne. -1) then
numpos = max (board(i, j), numpos)
ipos(board(i, j)) = i
jpos(board(i, j)) = j
end if
1000    continue
1010 continue

end

C-----------------------------------------------------------------------

subroutine nxtmov (board, n, imove, jmove, nmove)
implicit none

C     Find possible next moves. Prune and sort the moves according to
C     Warnsdorff's heuristic, keeping only those that have the minimum
C     number of legal following moves.

integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)

integer w1, w2, w3, w4, w5, w6, w7, w8
integer w
integer n1
integer pickw

call possib (board, n, imove, jmove, nmove)

n1 = n + 1
nmove(n1) = 1
call countf (board, n1, imove, jmove, nmove, w1)
nmove(n1) = 2
call countf (board, n1, imove, jmove, nmove, w2)
nmove(n1) = 3
call countf (board, n1, imove, jmove, nmove, w3)
nmove(n1) = 4
call countf (board, n1, imove, jmove, nmove, w4)
nmove(n1) = 5
call countf (board, n1, imove, jmove, nmove, w5)
nmove(n1) = 6
call countf (board, n1, imove, jmove, nmove, w6)
nmove(n1) = 7
call countf (board, n1, imove, jmove, nmove, w7)
nmove(n1) = 8
call countf (board, n1, imove, jmove, nmove, w8)

w = pickw (w1, w2, w3, w4, w5, w6, w7, w8)

if (w .eq. 0) then
call disabl (imove(1, n1), jmove(1, n1))
call disabl (imove(2, n1), jmove(2, n1))
call disabl (imove(3, n1), jmove(3, n1))
call disabl (imove(4, n1), jmove(4, n1))
call disabl (imove(5, n1), jmove(5, n1))
call disabl (imove(6, n1), jmove(6, n1))
call disabl (imove(7, n1), jmove(7, n1))
call disabl (imove(8, n1), jmove(8, n1))
else
if (w .ne. w1) call disabl (imove(1, n1), jmove(1, n1))
if (w .ne. w2) call disabl (imove(2, n1), jmove(2, n1))
if (w .ne. w3) call disabl (imove(3, n1), jmove(3, n1))
if (w .ne. w4) call disabl (imove(4, n1), jmove(4, n1))
if (w .ne. w5) call disabl (imove(5, n1), jmove(5, n1))
if (w .ne. w6) call disabl (imove(6, n1), jmove(6, n1))
if (w .ne. w7) call disabl (imove(7, n1), jmove(7, n1))
if (w .ne. w8) call disabl (imove(8, n1), jmove(8, n1))
end if

end

C-----------------------------------------------------------------------

subroutine countf (board, n, imove, jmove, nmove, w)
implicit none

C     Count the number of moves possible after an nth move.

integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)
integer w

logical goodmv
integer n1

if (goodmv (imove, nmove, n)) then
call mkmove (board, imove, jmove, nmove, n)
call possib (board, n, imove, jmove, nmove)
n1 = n + 1
w = 0
if (imove(1, n1) .ne. -1) w = w + 1
if (imove(2, n1) .ne. -1) w = w + 1
if (imove(3, n1) .ne. -1) w = w + 1
if (imove(4, n1) .ne. -1) w = w + 1
if (imove(5, n1) .ne. -1) w = w + 1
if (imove(6, n1) .ne. -1) w = w + 1
if (imove(7, n1) .ne. -1) w = w + 1
if (imove(8, n1) .ne. -1) w = w + 1
call unmove (board, imove, jmove, nmove, n)
else
C        The nth move itself is impossible.
w = 0
end if

end

C-----------------------------------------------------------------------

function pickw (w1, w2, w3, w4, w5, w6, w7, w8)
implicit none

C     From w1..w8, pick out the least nonzero value (or zero if they all
C     equal zero).

integer pickw
integer w1, w2, w3, w4, w5, w6, w7, w8

integer w
integer pickw1

w = 0
w = pickw1 (w, w1)
w = pickw1 (w, w2)
w = pickw1 (w, w3)
w = pickw1 (w, w4)
w = pickw1 (w, w5)
w = pickw1 (w, w6)
w = pickw1 (w, w7)
w = pickw1 (w, w8)

pickw = w

end

C-----------------------------------------------------------------------

function pickw1 (u, v)
implicit none

C     A small function used by pickw.

integer pickw1
integer u, v

if (v .eq. 0) then
pickw1 = u
else if (u .eq. 0) then
pickw1 = v
else
pickw1 = min (u, v)
end if

end

C-----------------------------------------------------------------------

subroutine possib (board, n, imove, jmove, nmove)
implicit none

C     Find moves that are possible from an nth-move position.

integer board(1:8,1:8)
integer n
integer imove(1:8,1:64)
integer jmove(1:8,1:64)
integer nmove(1:64)

integer i, j
integer n1

i = imove(nmove(n), n)
j = jmove(nmove(n), n)
n1 = n + 1
call trymov (board, i + 1, j + 2, imove(1, n1), jmove(1, n1))
call trymov (board, i + 2, j + 1, imove(2, n1), jmove(2, n1))
call trymov (board, i + 1, j - 2, imove(3, n1), jmove(3, n1))
call trymov (board, i + 2, j - 1, imove(4, n1), jmove(4, n1))
call trymov (board, i - 1, j + 2, imove(5, n1), jmove(5, n1))
call trymov (board, i - 2, j + 1, imove(6, n1), jmove(6, n1))
call trymov (board, i - 1, j - 2, imove(7, n1), jmove(7, n1))
call trymov (board, i - 2, j - 1, imove(8, n1), jmove(8, n1))

end

C-----------------------------------------------------------------------

subroutine trymov (board, i, j, imove, jmove)
implicit none

C     Try a move to square (i, j).

integer board(1:8,1:8)
integer i, j
integer imove, jmove

call disabl (imove, jmove)
if (1 .le. i .and. i .le. 8 .and. 1 .le. j .and. j .le. 8) then
if (board(i,j) .eq. -1) then
call enable (i, j, imove, jmove)
end if
end if

end

C-----------------------------------------------------------------------

subroutine enable (i, j, imove, jmove)
implicit none

C     Enable a potential move.

integer i, j
integer imove, jmove

imove = i
jmove = j

end

C-----------------------------------------------------------------------

subroutine disabl (imove, jmove)
implicit none

C     Disable a potential move.

integer imove, jmove

imove = -1
jmove = -1

end

C-----------------------------------------------------------------------

subroutine alg2ij (alg, i, j)
implicit none

C     Convert, for instance, 'c5' to i=3,j=5.

character*2 alg
integer i, j

if (alg(1:1) .eq. 'a') j = 1
if (alg(1:1) .eq. 'b') j = 2
if (alg(1:1) .eq. 'c') j = 3
if (alg(1:1) .eq. 'd') j = 4
if (alg(1:1) .eq. 'e') j = 5
if (alg(1:1) .eq. 'f') j = 6
if (alg(1:1) .eq. 'g') j = 7
if (alg(1:1) .eq. 'h') j = 8

if (alg(2:2) .eq. '1') i = 1
if (alg(2:2) .eq. '2') i = 2
if (alg(2:2) .eq. '3') i = 3
if (alg(2:2) .eq. '4') i = 4
if (alg(2:2) .eq. '5') i = 5
if (alg(2:2) .eq. '6') i = 6
if (alg(2:2) .eq. '7') i = 7
if (alg(2:2) .eq. '8') i = 8

end

C-----------------------------------------------------------------------

subroutine ij2alg (i, j, alg)
implicit none

C     Convert, for instance, i=3,j=5 to 'c5'.

integer i, j
character*2 alg

character alg1
character alg2

if (j .eq. 1) alg1 = 'a'
if (j .eq. 2) alg1 = 'b'
if (j .eq. 3) alg1 = 'c'
if (j .eq. 4) alg1 = 'd'
if (j .eq. 5) alg1 = 'e'
if (j .eq. 6) alg1 = 'f'
if (j .eq. 7) alg1 = 'g'
if (j .eq. 8) alg1 = 'h'

if (i .eq. 1) alg2 = '1'
if (i .eq. 2) alg2 = '2'
if (i .eq. 3) alg2 = '3'
if (i .eq. 4) alg2 = '4'
if (i .eq. 5) alg2 = '5'
if (i .eq. 6) alg2 = '6'
if (i .eq. 7) alg2 = '7'
if (i .eq. 8) alg2 = '8'

alg(1:1) = alg1
alg(2:2) = alg2

end

C-----------------------------------------------------------------------
```
Output:

\$ echo "c5 2 T" | ./knights_tour

``` Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 58 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 63 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 60 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h

Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 60 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 61 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 58 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h
```

### Fortran 95

Works with: gfortran version 11.2.1
Translation of: ATS
```!-----------------------------------------------------------------------
!
!     Find Knight’s Tours.
!
!     Using Warnsdorff’s heuristic, find multiple solutions.
!     Optionally accept only closed tours.
!
!     This program is migrated from my implementation for
!     ATS/Postiats. Unlike my FORTRAN 77 implementation (which simply
!     cannot do so), it uses a recursive call.
!
!     Compile with, for instance:
!
!         gfortran -O2 -g -std=f95 -o knights_tour knights_tour.f90
!
!     Usage examples:
!
!         One tour starting at a1, either open or closed:
!
!            echo "a1 1 F" | ./knights_tour
!
!         No more than 2000 closed tours starting at c5:
!
!            echo "c5 2000 T" | ./knights_tour
!
!-----------------------------------------------------------------------

program knights_tour
implicit none

character(len = 2) inp__alg
integer inp__istart
integer inp__jstart
integer inp__max_tours
logical inp__closed

read (*,*) inp__alg, inp__max_tours, inp__closed
call alg2ij (inp__alg, inp__istart, inp__jstart)
call main (inp__istart, inp__jstart, inp__max_tours, inp__closed)

contains

subroutine main (istart, jstart, max_tours, closed)
integer, intent(in) :: istart, jstart ! The starting position.
integer, intent(in) :: max_tours ! The max. no. of tours to print.
logical, intent(in) :: closed    ! Closed tours only?

integer board(1:8,1:8)
integer num_tours_printed

num_tours_printed = 0
call init_board (board)
call explore (board, 1, istart, jstart, max_tours, &
&        num_tours_printed, closed)
end subroutine main

recursive subroutine explore (board, n, i, j, max_tours, &
&                        num_tours_printed, closed)

! Recursively the space of 'Warnsdorffian' knight’s paths, looking
! for and printing complete tours.

integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(in) :: max_tours
integer, intent(inout) :: num_tours_printed
logical, intent(in) :: closed

integer imove(1:8)
integer jmove(1:8)
integer k

if (num_tours_printed < max_tours .and. n /= 0) then
if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
if (n == 63) then
call find_possible_moves (board, i, j, imove, jmove)
call try_last_move (board, n + 1, imove(1), jmove(1), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(2), jmove(2), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(3), jmove(3), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(4), jmove(4), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(5), jmove(5), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(6), jmove(6), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(7), jmove(7), &
&              num_tours_printed, closed)
call try_last_move (board, n + 1, imove(8), jmove(8), &
&              num_tours_printed, closed)
else
call find_next_moves (board, n, i, j, imove, jmove)
do k = 1, 8
if (is_good_move (imove(k), jmove(k))) then
!
! Here is the recursive call.
!
call explore (board, n + 1, imove(k), jmove(k), &
&        max_tours, num_tours_printed, closed)
end if
end do
end if
call unmove (board, i, j)
end if
end if
end subroutine explore

subroutine try_last_move (board, n, i, j, num_tours_printed, closed)
integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(inout) :: num_tours_printed
logical, intent(in) :: closed

integer ipos(1:64)
integer jpos(1:64)
integer numpos
integer idiff
integer jdiff

if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
if (.not. closed) then
num_tours_printed = num_tours_printed + 1
call print_tour (board, num_tours_printed)
else
call board2positions (board, ipos, jpos, numpos)
idiff = abs (i - ipos(1))
jdiff = abs (j - jpos(1))
if ((idiff == 1 .and. jdiff == 2) .or. &
(idiff == 2 .and. jdiff == 1)) then
num_tours_printed = num_tours_printed + 1
call print_tour (board, num_tours_printed)
end if
end if
call unmove (board, i, j)
end if
end subroutine try_last_move

subroutine init_board (board)

! Initialize a chessboard with empty squares.

integer, intent(out) :: board(1:8,1:8)

integer i, j

do j = 1, 8
do i = 1, 8
board(i, j) = -1
end do
end do
end subroutine init_board

subroutine mkmove (board, i, j, n)

! Fill a square with a move number.

integer, intent(inout) :: board(1:8, 1:8)
integer, intent(in) :: i, j
integer, intent(in) :: n

board(i, j) = n
end subroutine mkmove

subroutine unmove (board, i, j)

! Unmake a mkmove.

integer, intent(inout) :: board(1:8, 1:8)
integer, intent(in) :: i, j

board(i, j) = -1
end subroutine unmove

function is_good_move (i, j)
logical is_good_move
integer, intent(in) :: i, j

is_good_move = (i /= -1 .and. j /= -1)
end function is_good_move

subroutine print_tour (board, num_tours_printed)

! Print a knight's tour.

integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: num_tours_printed

write (*, '("Tour number ", I0)') num_tours_printed
call print_moves (board)
call print_board (board)
write (*, '()')
end subroutine print_tour

subroutine print_board (board)

! Print a chessboard with the move number in each square.

integer, intent(in) :: board(1:8,1:8)

integer i, j

do i = 8, 1, -1
write (*, '("    ", 8("+----"), "+")')
write (*, '(I2, " ", 8(" | ", I2), " | ")') &
i, (board(i, j), j = 1, 8)
end do
write (*, '("    ", 8("+----"), "+")')
write (*, '("   ", 8("    ", A1))') &
'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'

end subroutine print_board

subroutine print_moves (board)

! Print the moves of a knight's path, in algebraic notation.

integer, intent(in) :: board(1:8,1:8)

integer ipos(1:64)
integer jpos(1:64)
integer numpos
character(len = 2) alg(1:64)
integer columns(1:8)
integer k
integer m

character(len = 72) lines(1:8)

call board2positions (board, ipos, jpos, numpos)

! Convert the positions to algebraic notation.
do k = 1, numpos
call ij2alg (ipos(k), jpos(k), alg(k))
end do

! Fill lines with algebraic notations.
do m = 1, 8
columns(m) = 1
end do
m = 1
do k = 1, numpos
lines(m)(columns(m) : columns(m) + 1) = alg(k)(1:2)
columns(m) = columns(m) + 2
if (k /= numpos) then
lines(m)(columns(m) : columns(m) + 3) = " -> "
columns(m) = columns(m) + 4
else if (numpos == 64 .and. &
((abs (ipos(numpos) - ipos(1)) == 2 &
.and. abs (jpos(numpos) - jpos(1)) == 1) .or. &
((abs (ipos(numpos) - ipos(1)) == 1 &
.and. abs (jpos(numpos) - jpos(1)) == 2)))) then
lines(m)(columns(m) : columns(m) + 8) = " -> cycle"
columns(m) = columns(m) + 9
endif
if (mod (k, 8) == 0) m = m + 1
end do

! Print the lines that have stuff in them.
do m = 1, 8
if (columns(m) /= 1) then
write (*, '(A)') lines(m)(1 : columns(m) - 1)
end if
end do

end subroutine print_moves

function is_closed (board)

! Is a board a closed tour?

logical is_closed

integer board(1:8,1:8)
integer ipos(1:64)        ! The i-positions in order.
integer jpos(1:64)        ! The j-positions in order.
integer numpos            ! The number of positions so far.

call board2positions (board, ipos, jpos, numpos)

is_closed = (numpos == 64 .and. &
((abs (ipos(numpos) - ipos(1)) == 2 &
.and. abs (jpos(numpos) - jpos(1)) == 1) .or. &
((abs (ipos(numpos) - ipos(1)) == 1 &
.and. abs (jpos(numpos) - jpos(1)) == 2))))

end function is_closed

subroutine board2positions (board, ipos, jpos, numpos)

! Convert from a board to a list of board positions.

integer, intent(in) :: board(1:8,1:8)
integer, intent(out) :: ipos(1:64) ! The i-positions in order.
integer, intent(out) :: jpos(1:64) ! The j-positions in order.
integer, intent(out) :: numpos ! The number of positions so far.

integer i, j

numpos = 0
do i = 1, 8
do j = 1, 8
if (board(i, j) /= -1) then
numpos = max (board(i, j), numpos)
ipos(board(i, j)) = i
jpos(board(i, j)) = j
end if
end do
end do
end subroutine board2positions

subroutine find_next_moves (board, n, i, j, imove, jmove)

! Find possible next moves. Prune and sort the moves according to
! Warnsdorff's heuristic, keeping only those that have the minimum
! number of legal following moves.

integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(inout) :: imove(1:8)
integer, intent(inout) :: jmove(1:8)

integer w1, w2, w3, w4, w5, w6, w7, w8
integer w

call find_possible_moves (board, i, j, imove, jmove)

call count_following (board, n + 1, imove(1), jmove(1), w1)
call count_following (board, n + 1, imove(2), jmove(2), w2)
call count_following (board, n + 1, imove(3), jmove(3), w3)
call count_following (board, n + 1, imove(4), jmove(4), w4)
call count_following (board, n + 1, imove(5), jmove(5), w5)
call count_following (board, n + 1, imove(6), jmove(6), w6)
call count_following (board, n + 1, imove(7), jmove(7), w7)
call count_following (board, n + 1, imove(8), jmove(8), w8)

w = pick_w (w1, w2, w3, w4, w5, w6, w7, w8)

if (w == 0) then
call disable (imove(1), jmove(1))
call disable (imove(2), jmove(2))
call disable (imove(3), jmove(3))
call disable (imove(4), jmove(4))
call disable (imove(5), jmove(5))
call disable (imove(6), jmove(6))
call disable (imove(7), jmove(7))
call disable (imove(8), jmove(8))
else
if (w /= w1) call disable (imove(1), jmove(1))
if (w /= w2) call disable (imove(2), jmove(2))
if (w /= w3) call disable (imove(3), jmove(3))
if (w /= w4) call disable (imove(4), jmove(4))
if (w /= w5) call disable (imove(5), jmove(5))
if (w /= w6) call disable (imove(6), jmove(6))
if (w /= w7) call disable (imove(7), jmove(7))
if (w /= w8) call disable (imove(8), jmove(8))
end if

end subroutine find_next_moves

subroutine count_following (board, n, i, j, w)

! Count the number of moves possible after an nth move.

integer, intent(inout) :: board(1:8,1:8)
integer, intent(in) :: n
integer, intent(in) :: i, j
integer, intent(out) :: w

integer imove(1:8)
integer jmove(1:8)

if (is_good_move (i, j)) then
call mkmove (board, i, j, n)
call find_possible_moves (board, i, j, imove, jmove)
w = 0
if (is_good_move (imove(1), jmove(1))) w = w + 1
if (is_good_move (imove(2), jmove(2))) w = w + 1
if (is_good_move (imove(3), jmove(3))) w = w + 1
if (is_good_move (imove(4), jmove(4))) w = w + 1
if (is_good_move (imove(5), jmove(5))) w = w + 1
if (is_good_move (imove(6), jmove(6))) w = w + 1
if (is_good_move (imove(7), jmove(7))) w = w + 1
if (is_good_move (imove(8), jmove(8))) w = w + 1
call unmove (board, i, j)
else
! The nth move itself is impossible.
w = 0
end if

end subroutine count_following

function pick_w (w1, w2, w3, w4, w5, w6, w7, w8) result (w)

! From w1..w8, pick out the least nonzero value (or zero if they
! all equal zero).

integer, intent(in) :: w1, w2, w3, w4, w5, w6, w7, w8
integer w

w = 0
w = pick_w1 (w, w1)
w = pick_w1 (w, w2)
w = pick_w1 (w, w3)
w = pick_w1 (w, w4)
w = pick_w1 (w, w5)
w = pick_w1 (w, w6)
w = pick_w1 (w, w7)
w = pick_w1 (w, w8)
end function pick_w

function pick_w1 (u, v)

! A small function used by pick_w.

integer pick_w1
integer, intent(in) :: u, v

if (v == 0) then
pick_w1 = u
else if (u == 0) then
pick_w1 = v
else
pick_w1 = min (u, v)
end if
end function pick_w1

subroutine find_possible_moves (board, i, j, imove, jmove)

! Find moves that are possible from a position.

integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j
integer, intent(out) :: imove(1:8)
integer, intent(out) :: jmove(1:8)

call trymov (board, i + 1, j + 2, imove(1), jmove(1))
call trymov (board, i + 2, j + 1, imove(2), jmove(2))
call trymov (board, i + 1, j - 2, imove(3), jmove(3))
call trymov (board, i + 2, j - 1, imove(4), jmove(4))
call trymov (board, i - 1, j + 2, imove(5), jmove(5))
call trymov (board, i - 2, j + 1, imove(6), jmove(6))
call trymov (board, i - 1, j - 2, imove(7), jmove(7))
call trymov (board, i - 2, j - 1, imove(8), jmove(8))
end subroutine find_possible_moves

subroutine trymov (board, i, j, imove, jmove)

! Try a move to square (i, j).

integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j
integer, intent(inout) :: imove, jmove

call disable (imove, jmove)
if (1 <= i .and. i <= 8 .and. 1 <= j .and. j <= 8) then
if (square_is_empty (board, i, j)) then
call enable (i, j, imove, jmove)
end if
end if

end subroutine trymov

function square_is_empty (board, i, j)
logical square_is_empty
integer, intent(in) :: board(1:8,1:8)
integer, intent(in) :: i, j

square_is_empty = (board(i, j) == -1)
end function square_is_empty

subroutine enable (i, j, imove, jmove)

! Enable a potential move.

integer, intent(in) :: i, j
integer, intent(inout) :: imove, jmove

imove = i
jmove = j
end subroutine enable

subroutine disable (imove, jmove)

! Disable a potential move.

integer, intent(out) :: imove, jmove

imove = -1
jmove = -1
end subroutine disable

subroutine alg2ij (alg, i, j)

! Convert, for instance, 'c5' to i=3,j=5.

character(len = 2), intent(in) :: alg
integer, intent(out) :: i, j

if (alg(1:1) == 'a') j = 1
if (alg(1:1) == 'b') j = 2
if (alg(1:1) == 'c') j = 3
if (alg(1:1) == 'd') j = 4
if (alg(1:1) == 'e') j = 5
if (alg(1:1) == 'f') j = 6
if (alg(1:1) == 'g') j = 7
if (alg(1:1) == 'h') j = 8

if (alg(2:2) == '1') i = 1
if (alg(2:2) == '2') i = 2
if (alg(2:2) == '3') i = 3
if (alg(2:2) == '4') i = 4
if (alg(2:2) == '5') i = 5
if (alg(2:2) == '6') i = 6
if (alg(2:2) == '7') i = 7
if (alg(2:2) == '8') i = 8

end subroutine alg2ij

subroutine ij2alg (i, j, alg)

! Convert, for instance, i=3,j=5 to 'c5'.

integer, intent(in) :: i, j
character(len = 2), intent(out) :: alg

character alg1
character alg2

if (j == 1) alg1 = 'a'
if (j == 2) alg1 = 'b'
if (j == 3) alg1 = 'c'
if (j == 4) alg1 = 'd'
if (j == 5) alg1 = 'e'
if (j == 6) alg1 = 'f'
if (j == 7) alg1 = 'g'
if (j == 8) alg1 = 'h'

if (i == 1) alg2 = '1'
if (i == 2) alg2 = '2'
if (i == 3) alg2 = '3'
if (i == 4) alg2 = '4'
if (i == 5) alg2 = '5'
if (i == 6) alg2 = '6'
if (i == 7) alg2 = '7'
if (i == 8) alg2 = '8'

alg(1:1) = alg1
alg(2:2) = alg2

end subroutine ij2alg

end program

!-----------------------------------------------------------------------
```
Output:

\$ echo "c5 2 T" | ./knights_tour

```Tour number 1
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> e8 -> d6 -> b5 -> d4 -> f5 -> g7 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 58 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 63 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 60 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 61 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h

Tour number 2
c5 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> h3 ->
g1 -> e2 -> c1 -> a2 -> b4 -> d3 -> e1 -> g2 ->
h4 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f3 -> h2 -> f1 ->
g3 -> h1 -> f2 -> e4 -> c3 -> a4 -> b2 -> d1 ->
e3 -> g4 -> h6 -> g8 -> f6 -> h5 -> f4 -> d5 ->
e7 -> c8 -> a7 -> c6 -> e5 -> c4 -> b6 -> a8 ->
c7 -> b5 -> d6 -> e8 -> g7 -> f5 -> d4 -> e6 -> cycle
+----+----+----+----+----+----+----+----+
8  | 56 |  3 | 50 | 21 | 60 |  5 | 44 | 19 |
+----+----+----+----+----+----+----+----+
7  | 51 | 22 | 57 |  4 | 49 | 20 | 61 |  6 |
+----+----+----+----+----+----+----+----+
6  |  2 | 55 | 52 | 59 | 64 | 45 | 18 | 43 |
+----+----+----+----+----+----+----+----+
5  | 23 | 58 |  1 | 48 | 53 | 62 |  7 | 46 |
+----+----+----+----+----+----+----+----+
4  | 38 | 13 | 54 | 63 | 36 | 47 | 42 | 17 |
+----+----+----+----+----+----+----+----+
3  | 27 | 24 | 37 | 14 | 41 | 30 | 33 |  8 |
+----+----+----+----+----+----+----+----+
2  | 12 | 39 | 26 | 29 | 10 | 35 | 16 | 31 |
+----+----+----+----+----+----+----+----+
1  | 25 | 28 | 11 | 40 | 15 | 32 |  9 | 34 |
+----+----+----+----+----+----+----+----+
a    b    c    d    e    f    g    h
```

### Fortran 2008

Works with: gfortran version 11.2.1

(This one is not a translation of my ATS implementation. I wrote it earlier.)

```!!!
!!! Find a Knight’s Tour.
!!!
!!! Use Warnsdorff’s heuristic, but write the program so it should not
!!! be able to terminate unsuccessfully.
!!!

module knights_tour
use, intrinsic :: iso_fortran_env, only: output_unit, error_unit

implicit none
private

public :: find_a_knights_tour
public :: notation_is_a_square

integer, parameter :: number_of_ranks = 8
integer, parameter :: number_of_files = 8
integer, parameter :: number_of_squares = number_of_ranks * number_of_files

! ‘Algebraic’ chess notation.
character, parameter :: rank_notation(1:8) = (/ '1', '2', '3', '4', '5', '6', '7', '8' /)
character, parameter :: file_notation(1:8) = (/ 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h' /)

type :: board_square_t
! Squares are represented by their algebraic notation.
character(2) :: algebraic_notation
contains
procedure, pass :: output => board_square_t_output
procedure, pass :: knight_moves => board_square_t_knight_moves
procedure, pass :: equal => board_square_t_equal
generic :: operator(==) => equal
end type board_square_t

type :: knight_moves_t
integer :: number_of_squares
type(board_square_t) :: squares(1:8)
end type knight_moves_t

type :: path_t
integer :: length
type(board_square_t) :: squares(1:number_of_squares)
contains
procedure, pass :: output => path_t_output
end type path_t

contains

pure function notation_is_a_square (notation) result (bool)
character(*), intent(in) :: notation
logical :: bool

integer :: length
integer :: rank_no
integer :: file_no

length = len_trim (notation)
if (length /= 2) then
bool = .false.
else
rank_no = findloc (rank_notation, notation(2:2), 1)
file_no = findloc (file_notation, notation(1:1), 1)
bool = (1 <= rank_no .and. rank_no <= number_of_ranks)         &
&   .and. (1 <= file_no .and. file_no <= number_of_files)
end if
end function notation_is_a_square

subroutine path_t_output (path, unit)
!
! Print a path in algebraic notation.
!
class(path_t), intent(in) :: path
integer, intent(in) :: unit

integer :: moves_counter
integer :: i

moves_counter = 1
if (1 <= path%length) then
call path%squares(1)%output(unit)
do i = 2, path%length
if (moves_counter == 8) then
write (unit, '(" ->")', advance = 'yes')
moves_counter = 1
else
write (unit, '(" -> ")', advance = 'no')
moves_counter = moves_counter + 1
end if
call path%squares(i)%output(unit)
end do
end if
write (output_unit, '()')
end subroutine path_t_output

subroutine board_square_t_output (square, unit)
!
! Print a square in algebraic notation.
!
class(board_square_t), intent(in) :: square
integer, intent(in) :: unit

write (unit, '(A2)', advance = 'no') square%algebraic_notation
end subroutine board_square_t_output

elemental function board_square_t_equal (p, q) result (bool)
class(board_square_t), intent(in) :: p, q
logical :: bool

bool = (p%algebraic_notation == q%algebraic_notation)
end function board_square_t_equal

pure function board_square_t_knight_moves (square) result (moves)
!
! Return all possible moves of a knight from a given square.
!
class(board_square_t), intent(in) :: square
type(knight_moves_t) :: moves

integer, parameter :: rank_stride(1:number_of_ranks) = (/ +1, +2, +1, +2, -1, -2, -1, -2 /)
integer, parameter :: file_stride(1:number_of_files) = (/ +2, +1, -2, -1, +2, +1, -2, -1 /)

integer :: rank_no, file_no
integer :: new_rank_no, new_file_no
integer :: i
character(2) :: notation

rank_no = findloc (rank_notation, square%algebraic_notation(2:2), 1)
file_no = findloc (file_notation, square%algebraic_notation(1:1), 1)

moves%number_of_squares = 0
do i = 1, 8
new_rank_no = rank_no + rank_stride(i)
new_file_no = file_no + file_stride(i)
if (1 <= new_rank_no                           &
& .and. new_rank_no <= number_of_ranks    &
& .and. 1 <= new_file_no                  &
& .and. new_file_no <= number_of_files) then
moves%number_of_squares = moves%number_of_squares + 1
notation(2:2) = rank_notation(new_rank_no)
notation(1:1) = file_notation(new_file_no)
moves%squares(moves%number_of_squares) = board_square_t (notation)
end if
end do
end function board_square_t_knight_moves

pure function unvisited_knight_moves (path) result (moves)
!
! Return moves of a knight from a given square, but only those
! that have not been visited already.
!
class(path_t), intent(in) :: path
type(knight_moves_t) :: moves

type(knight_moves_t) :: all_moves
integer :: i

all_moves = path%squares(path%length)%knight_moves()
moves%number_of_squares = 0
do i = 1, all_moves%number_of_squares
if (all (.not. all_moves%squares(i) == path%squares(1:path%length))) then
moves%number_of_squares = moves%number_of_squares + 1
moves%squares(moves%number_of_squares) = all_moves%squares(i)
end if
end do
end function unvisited_knight_moves

pure function potential_knight_moves (path) result (moves)
!
! Return moves of a knight from a given square, but only those
! that are unvisited, and from which another unvisited move can be
!
! Sort the returned moves in nondecreasing order of the number of
! possible moves after the first. (This is how we implement
! Warnsdorff’s heuristic.)
!
class(path_t), intent(in) :: path
type(knight_moves_t) :: moves

type(knight_moves_t) :: unvisited_moves
type(knight_moves_t) :: next_moves
type(path_t) :: next_path
type(board_square_t) :: unpruned_squares(1:8)
integer :: warnsdorff_numbers(1:8)
integer :: number_of_unpruned_squares
integer :: i

if (path%length == number_of_squares - 1) then
!
! There is only one square left on the board. Either the knight
! can reach it or it cannot.
!
moves = unvisited_knight_moves (path)
else
!
! Use Warnsdorff’s heuristic: return unvisited moves, but try
! first those with the least number of possible moves following
! it.
!
! If the number of possible moves following is zero, prune the
! move, because it is a dead end.
!
number_of_unpruned_squares = 0
unvisited_moves = unvisited_knight_moves (path)
do i = 1, unvisited_moves%number_of_squares
next_path%length = path%length + 1
next_path%squares(1:path%length) = path%squares(1:path%length)
next_path%squares(next_path%length) = unvisited_moves%squares(i)

next_moves = unvisited_knight_moves (next_path)

if (next_moves%number_of_squares /= 0) then
number_of_unpruned_squares = number_of_unpruned_squares + 1
unpruned_squares(number_of_unpruned_squares) = unvisited_moves%squares(i)
warnsdorff_numbers(number_of_unpruned_squares) = next_moves%number_of_squares
end if
end do

! In-place insertion sort of the unpruned squares.
block
type(board_square_t) :: square
integer :: w_number
integer :: i, j

i = 2
do while (i <= number_of_unpruned_squares)
square = unpruned_squares(i)
w_number = warnsdorff_numbers(i)
j = i - 1
do while (1 <= j .and. w_number < warnsdorff_numbers(j))
unpruned_squares(j + 1) = unpruned_squares(j)
warnsdorff_numbers(j + 1) = warnsdorff_numbers(j)
j = j - 1
end do
unpruned_squares(j + 1) = square
warnsdorff_numbers(j + 1) = w_number
i = i + 1
end do
end block

moves%number_of_squares = number_of_unpruned_squares
moves%squares(1:number_of_unpruned_squares) = &
& unpruned_squares(1:number_of_unpruned_squares)
end if
end function potential_knight_moves

subroutine find_a_knights_tour (starting_square)
!
! Find and print a full knight’s tour.
!
character(2), intent(in) :: starting_square

type(path_t) :: path

path%length = 1
path%squares(1) = board_square_t (starting_square)
path = try_paths (path)
if (path%length /= 0) then
call path%output(output_unit)
else
write (error_unit, '("The program terminated without finding a solution.")')
write (error_unit, '("This is supposed to be impossible for an 8-by-8 board.")')
write (error_unit, '("The program is wrong.")')
error stop
end if

contains

recursive function try_paths (path) result (solution)
!
! Recursively try all possible paths, but using Warnsdorff’s
! heuristic to speed up the search.
!
class(path_t), intent(in) :: path
type(path_t) :: solution

type(path_t) :: new_path
type(knight_moves_t) :: moves
integer :: i

if (path%length == number_of_squares) then
solution = path
else
solution%length = 0

moves = potential_knight_moves (path)

if (moves%number_of_squares /= 0) then
new_path%length = path%length + 1
new_path%squares(1:path%length) = path%squares(1:path%length)

i = 1
do while (solution%length == 0 .and. i <= moves%number_of_squares)
new_path%squares(new_path%length) = moves%squares(i)
solution = try_paths (new_path)
i = i + 1
end do
end if
end if
end function try_paths

end subroutine find_a_knights_tour

end module knights_tour

program knights_tour_main
use, intrinsic :: iso_fortran_env, only: output_unit
use, non_intrinsic :: knights_tour
implicit none

character(200) :: arg
integer :: arg_count
integer :: i

arg_count = command_argument_count ()
do i = 1, arg_count
call get_command_argument (i, arg)
arg = adjustl (arg)
if (1 < i) write (output_unit, '()')
if (notation_is_a_square (arg)) then
call find_a_knights_tour (arg)
else
write (output_unit, '("This is not algebraic notation: ", A)') arg
end if
end do
end program knights_tour_main
```

\$ ./knights_tour a1 b2 c3

```a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> g4 ->
h6 -> g8 -> e7 -> c8 -> a7 -> b5 -> c7 -> a8 ->
b6 -> a4 -> b2 -> d1 -> f2 -> h1 -> g3 -> h5 ->
g7 -> e8 -> f6 -> h7 -> f8 -> d7 -> b8 -> a6 ->
b4 -> a2 -> c3 -> d5 -> e3 -> f5 -> h4 -> g2 ->
e1 -> f3 -> g1 -> h3 -> g5 -> e4 -> d6 -> c4 ->
a5 -> b7 -> d8 -> f7 -> h8 -> g6 -> e5 -> c6 ->
d4 -> e6 -> f4 -> e2 -> c1 -> d3 -> c5 -> b3

b2 -> a4 -> b6 -> a8 -> c7 -> e8 -> g7 -> h5 ->
g3 -> h1 -> f2 -> d1 -> c3 -> a2 -> c1 -> e2 ->
g1 -> h3 -> f4 -> g2 -> h4 -> g6 -> h8 -> f7 ->
d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> e1 -> d3 ->
b4 -> a6 -> b8 -> d7 -> f8 -> h7 -> g5 -> e6 ->
c5 -> e4 -> f6 -> g8 -> h6 -> g4 -> h2 -> f1 ->
d2 -> b1 -> a3 -> c4 -> e5 -> f3 -> d4 -> b5 ->
d6 -> c8 -> a7 -> c6 -> e7 -> f5 -> e3 -> d5

c3 -> a2 -> c1 -> e2 -> g1 -> h3 -> g5 -> h7 ->
f8 -> g6 -> h8 -> f7 -> d8 -> b7 -> a5 -> b3 ->
a1 -> c2 -> a3 -> b1 -> d2 -> f1 -> h2 -> f3 ->
h4 -> g2 -> e1 -> d3 -> f4 -> h5 -> g7 -> e8 ->
f6 -> g8 -> h6 -> g4 -> e5 -> d7 -> b8 -> a6 ->
b4 -> c6 -> a7 -> c8 -> e7 -> d5 -> b6 -> a8 ->
c7 -> e6 -> c5 -> a4 -> b2 -> c4 -> e3 -> d1 ->
f2 -> h1 -> g3 -> e4 -> d6 -> f5 -> d4 -> b5```

## Go

### Warnsdorf's rule

```package main

import (
"fmt"
"math/rand"
"time"
)

// input, 0-based start position
const startRow = 0
const startCol = 0

func main() {
rand.Seed(time.Now().Unix())
for !knightTour() {
}
}

var moves = []struct{ dr, dc int }{
{2, 1},
{2, -1},
{1, 2},
{1, -2},
{-1, 2},
{-1, -2},
{-2, 1},
{-2, -1},
}

// Attempt knight tour starting at startRow, startCol using Warnsdorff's rule
// and random tie breaking.  If a tour is found, print it and return true.
// Otherwise no backtracking, just return false.
func knightTour() bool {
// 8x8 board.  squares hold 1-based visit order.  0 means unvisited.
board := make([][]int, 8)
for i := range board {
board[i] = make([]int, 8)
}
r := startRow
c := startCol
board[r][c] = 1 // first move
for move := 2; move <= 64; move++ {
minNext := 8
var mr, mc, nm int
candidateMoves:
for _, cm := range moves {
cr := r + cm.dr
if cr < 0 || cr >= 8 { // off board
continue
}
cc := c + cm.dc
if cc < 0 || cc >= 8 { // off board
continue
}
if board[cr][cc] > 0 { // already visited
continue
}
// cr, cc candidate legal move.
p := 0 // count possible next moves.
for _, m2 := range moves {
r2 := cr + m2.dr
if r2 < 0 || r2 >= 8 {
continue
}
c2 := cc + m2.dc
if c2 < 0 || c2 >= 8 {
continue
}
if board[r2][c2] > 0 {
continue
}
p++
if p > minNext { // bail out as soon as it's eliminated
continue candidateMoves
}
}
if p < minNext { // it's better.  keep it.
minNext = p // new min possible next moves
nm = 1      // number of candidates with this p
mr = cr     // best candidate move
mc = cc
continue
}
// it ties for best so far.
// keep it with probability 1/(number of tying moves)
nm++                    // number of tying moves
if rand.Intn(nm) == 0 { // one chance to keep it
mr = cr
mc = cc
}
}
if nm == 0 { // no legal move
return false
}
// make selected move
r = mr
c = mc
board[r][c] = move
}
// tour complete.  print board.
for _, r := range board {
for _, m := range r {
fmt.Printf("%3d", m)
}
fmt.Println()
}
return true
}
```
Output:
```  1  4 39 20 23  6 63 58
40 19  2  5 62 57 22  7
3 38 41 48 21 24 59 64
18 43 32 37 56 61  8 25
31 14 47 42 49 36 53 60
46 17 44 33 52 55 26  9
13 30 15 50 11 28 35 54
16 45 12 29 34 51 10 27
```

### Ant colony

```/* Adapted from "Enumerating Knight's Tours using an Ant Colony Algorithm"
by Philip Hingston and Graham Kendal,
PDF at http://www.cs.nott.ac.uk/~gxk/papers/cec05knights.pdf. */

package main

import (
"fmt"
"math/rand"
"sync"
"time"
)

const boardSize = 8
const nSquares = boardSize * boardSize
const completeTour = nSquares - 1

// task input: starting square.  These are 1 based, but otherwise 0 based
// row and column numbers are used througout the program.
const rStart = 2
const cStart = 3

// pheromone representation read by ants
var tNet = make([]float64, nSquares*8)

// row, col deltas of legal moves
var drc = [][]int{{1, 2}, {2, 1}, {2, -1}, {1, -2},
{-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}}

// get square reached by following edge k from square (r, c)
func dest(r, c, k int) (int, int, bool) {
r += drc[k][0]
c += drc[k][1]
return r, c, r >= 0 && r < boardSize && c >= 0 && c < boardSize
}

// struct represents a pheromone amount associated with a move
type rckt struct {
r, c, k int
t       float64
}

func main() {
fmt.Println("Starting square:  row", rStart, "column", cStart)
// initialize board
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
for k := 0; k < 8; k++ {
if _, _, ok := dest(r, c, k); ok {
tNet[(r*boardSize+c)*8+k] = 1e-6
}
}
}
}

// waitGroups for ant release clockwork
var start, reset sync.WaitGroup
// channel for ants to return tours with pheremone updates
tch := make(chan []rckt)

// create an ant for each square
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
go ant(r, c, &start, &reset, tch)
}
}

// accumulator for new pheromone amounts
tNew := make([]float64, nSquares*8)

// each iteration is a "cycle" as described in the paper
for {
// evaporate pheromones
for i := range tNet {
tNet[i] *= .75
}

reset.Add(nSquares) // number of ants to release
start.Done()        // release them
reset.Wait()        // wait for them to begin searching
start.Add(1)        // reset start signal for next cycle

// gather tours from ants
for i := 0; i < nSquares; i++ {
tour := <-tch
// watch for a complete tour from the specified starting square
if len(tour) == completeTour &&
tour[0].r == rStart-1 && tour[0].c == cStart-1 {

// task output:  move sequence in a grid.
seq := make([]int, nSquares)
for i, sq := range tour {
seq[sq.r*boardSize+sq.c] = i + 1
}
last := tour[len(tour)-1]
r, c, _ := dest(last.r, last.c, last.k)
seq[r*boardSize+c] = nSquares
fmt.Println("Move sequence:")
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
fmt.Printf(" %3d", seq[r*boardSize+c])
}
fmt.Println()
}
return // task only requires finding a single tour
}
// accumulate pheromone amounts from all ants
for _, move := range tour {
tNew[(move.r*boardSize+move.c)*8+move.k] += move.t
}
}

// update pheromone amounts on network, reset accumulator
for i, tn := range tNew {
tNet[i] += tn
tNew[i] = 0
}
}
}

type square struct {
r, c int
}

func ant(r, c int, start, reset *sync.WaitGroup, tourCh chan []rckt) {
rnd := rand.New(rand.NewSource(time.Now().UnixNano()))
tabu := make([]square, nSquares)
moves := make([]rckt, nSquares)
unexp := make([]rckt, 8)
tabu[0].r = r
tabu[0].c = c

for {
// cycle initialization
moves = moves[:0]
tabu = tabu[:1]
r := tabu[0].r
c := tabu[0].c

// wait for start signal
start.Wait()
reset.Done()

for {
// choose next move
unexp = unexp[:0]
var tSum float64
findU:
for k := 0; k < 8; k++ {
dr, dc, ok := dest(r, c, k)
if !ok {
continue
}
for _, t := range tabu {
if t.r == dr && t.c == dc {
continue findU
}
}
tk := tNet[(r*boardSize+c)*8+k]
tSum += tk
// note:  dest r, c stored here
unexp = append(unexp, rckt{dr, dc, k, tk})
}
if len(unexp) == 0 {
break // no moves
}
rn := rnd.Float64() * tSum
var move rckt
for _, move = range unexp {
if rn <= move.t {
break
}
rn -= move.t
}

// move to new square
move.r, r = r, move.r
move.c, c = c, move.c
tabu = append(tabu, square{r, c})
moves = append(moves, move)
}

// compute pheromone amount to leave
for i := range moves {
moves[i].t = float64(len(moves)-i) / float64(completeTour-i)
}

// return tour found for this cycle
tourCh <- moves
}
}
```

Output:

```Starting square:  row 2 column 3
Move sequence:
64  33  36   3  54  49  38  51
35   4   1  30  37  52  55  48
32  63  34  53   2  47  50  39
5  18  31  46  29  20  13  56
62  27  44  19  14  11  40  21
17   6  15  28  45  22  57  12
26  61   8  43  24  59  10  41
7  16  25  60   9  42  23  58
```

```import Data.Bifunctor (bimap)
import Data.Char (chr, ord)
import Data.List (intercalate, minimumBy, sort, (\\))
import Data.Ord (comparing)

---------------------- KNIGHT'S TOUR ---------------------

type Square = (Int, Int)

knightTour :: [Square] -> [Square]
knightTour moves
| null possibilities = reverse moves
| otherwise = knightTour \$ newSquare : moves
where
newSquare =
minimumBy
(comparing (length . findMoves))
possibilities
possibilities = findMoves \$ head moves
findMoves = (\\ moves) . knightOptions

knightOptions :: Square -> [Square]
knightOptions (x, y) =
knightMoves >>= go . bimap (+ x) (+ y)
where
go move
| uncurry (&&) (both onBoard move) = [move]
| otherwise = []

knightMoves :: [(Int, Int)]
knightMoves =
((>>=) <*> (\deltas n -> deltas >>= go n)) [1, 2, -1, -2]
where
go i x
| abs i /= abs x = [(i, x)]
| otherwise = []

onBoard :: Int -> Bool
onBoard = (&&) . (0 <) <*> (9 >)

both :: (a -> b) -> (a,  a) -> (b,  b)
both = join bimap

--------------------------- TEST -------------------------
startPoint :: String
startPoint = "e5"

algebraic :: (Int, Int) -> String
algebraic (x, y) = [chr (x + 96), chr (y + 48)]

main :: IO ()
main =
printTour \$
algebraic
<\$> knightTour
[(\[x, y] -> (ord x - 96, ord y - 48)) startPoint]
where
printTour [] = return ()
printTour tour = do
putStrLn \$ intercalate " -> " \$ take 8 tour
printTour \$ drop 8 tour
```
Output:
```e5 -> f7 -> h8 -> g6 -> h4 -> g2 -> e1 -> f3
g1 -> h3 -> g5 -> h7 -> f8 -> d7 -> b8 -> a6
b4 -> a2 -> c1 -> d3 -> b2 -> d1 -> f2 -> h1
g3 -> h5 -> g7 -> e8 -> f6 -> g8 -> h6 -> g4
h2 -> f1 -> e3 -> f5 -> e7 -> c8 -> a7 -> c6
d8 -> b7 -> a5 -> b3 -> a1 -> c2 -> d4 -> e2
f4 -> e6 -> c5 -> a4 -> b6 -> a8 -> c7 -> d5
c3 -> e4 -> d6 -> b5 -> a3 -> b1 -> d2 -> c4```

## Icon and Unicon

This implements Warnsdorff's algorithm using unordered sets.

• The board must be square (it has only been tested on 8x8 and 7x7). Currently the maximum size board (due to square notation) is 26x26.
• Tie breaking is selectable with 3 variants supplied (first in list, random, and Roth's distance heuristic).
• A debug log can be generated showing the moves and choices considered for tie breaking.

The algorithm doesn't always generate a complete tour.

```link printf

procedure main(A)
ShowTour(KnightsTour(Board(8)))
end

procedure KnightsTour(B,sq,tbrk,debug)  #: Warnsdorff’s algorithm

/B := Board(8)                          # create 8x8 board if none given
/sq := ?B.files || ?B.ranks             # random initial position (default)
sq2fr(sq,B)                             # validate initial sq
if type(tbrk) == "procedure" then
B.tiebreak := tbrk                   # override tie-breaker
if \debug then write("Debug log : move#, move : (accessibility) choices")

choices := []                           # setup to track moves and choices
every (movesto := table())[k := key(B.movesto)] := copy(B.movesto[k])

B.tour := []                            # new tour
repeat {
put(B.tour,sq)                       # record move

ac := 9                              # accessibility counter > maximum
while get(choices)                   # empty choices for tiebreak
every delete(movesto[nextsq := !movesto[sq]],sq) do {  # make sq unavailable
if ac >:= *movesto[nextsq] then   # reset to lower accessibility count
while get(choices)             # . re-empty choices
if ac = *movesto[nextsq] then
put(choices,nextsq)            # keep least accessible sq and any ties
}

if \debug then {                     # move#, move, (accessibility), choices
writes(sprintf("%d. %s : (%d) ",*B.tour,sq,ac))
every writes(" ",!choices|"\n")
}
sq := B.tiebreak(choices,B) | break  # choose next sq until out of choices
}
return B
end

procedure RandomTieBreaker(S,B)                   # random choice
return ?S
end

procedure FirstTieBreaker(S,B)                    # first one in the list
return !S
end

procedure RothTieBreaker(S,B)                    # furthest from the center
if *S = 0 then fail                              # must fail if []
every fr := sq2fr(s := !S,B) do {
d := sqrt(abs(fr[1]-1 - (B.N-1)*0.5)^2 + abs(fr[2]-1 - (B.N-1)*0.5)^2)
if (/md := d) | ( md >:= d) then msq := s     # save sq
}
return msq
end

record board(N,ranks,files,movesto,tiebreak,tour)  # structure for board

procedure Board(N)                      #: create board
N := *&lcase >=( 0 < integer(N)) | stop("N=",image(N)," is out of range.")
B := board(N,[],&lcase[1+:N],table(),RandomTieBreaker)       # setup
every put(B.ranks,N to 1 by -1)                              # add rank #s
every sq := !B.files || !B.ranks do                          # for each sq add
every insert(B.movesto[sq] := set(), KnightMoves(sq,B))   # moves to next sq
return B
end

procedure sq2fr(sq,B)                   #: return numeric file & rank
f := find(sq[1],B.files)                | runerr(205,sq)
r := integer(B.ranks[sq[2:0]])          | runerr(205,sq)
return [f,r]
end

procedure KnightMoves(sq,B)         #: generate all Kn accessible moves from sq
fr := sq2fr(sq,B)
every ( i := -2|-1|1|2 ) & ( j := -2|-1|1|2 ) do
if (abs(i)~=abs(j)) & (0<(ri:=fr[2]+i)<=B.N) & (0<(fj:=fr[1]+j)<=B.N) then
suspend B.files[fj]||B.ranks[ri]
end

procedure ShowTour(B)                   #: show the tour
write("Board size = ",B.N)
write("Tour length = ",*B.tour)
write("Tie Breaker = ",image(B.tiebreak))

every !(squares := list(B.N)) := list(B.N,"-")
every fr := sq2fr(B.tour[m := 1 to *B.tour],B) do
squares[fr[2],fr[1]] := m

every (hdr1 := "     ") ||:= right(!B.files,3)
every (hdr2 := "    +") ||:= repl((1 to B.N,"-"),3) | "-+"

every write(hdr1|hdr2)
every r := 1 to B.N do {
writes(right(B.ranks[r],3)," |")
every writes(right(squares[r,f := 1 to B.N],3))
write(" |",right(B.ranks[r],3))
}
every write(hdr2|hdr1|&null)
end
```

The following can be used when debugging to validate the board structure and to image the available moves on the board.

```procedure DumpBoard(B)  #: Dump Board internals
write("Board size=",B.N)
write("Available Moves at start of tour:", ImageMovesTo(B.movesto))
end

procedure ImageMovesTo(movesto)  #: image of available moves
every put(K := [],key(movesto))
every (s := "\n") ||:= (k := !sort(K)) || " : " do
every s ||:= " " || (!sort(movesto[k])|"\n")
return s
end
```

Sample output:

```Board size = 8
Tour length = 64
Tie Breaker = procedure RandomTieBreaker
a  b  c  d  e  f  g  h
+-------------------------+
8 | 53 10 29 26 55 12 31 16 |  8
7 | 28 25 54 11 30 15 48 13 |  7
6 |  9 52 27 62 47 56 17 32 |  6
5 | 24 61 38 51 36 45 14 49 |  5
4 | 39  8 63 46 57 50 33 18 |  4
3 | 64 23 60 37 42 35 44  3 |  3
2 |  7 40 21 58  5  2 19 34 |  2
1 | 22 59  6 41 20 43  4  1 |  1
+-------------------------+
a  b  c  d  e  f  g  h```

Two 7x7 boards:

```Board size = 7
Tour length = 33
Tie Breaker = procedure RandomTieBreaker
a  b  c  d  e  f  g
+----------------------+
7 | 33  4 15  - 29  6 17 |  7
6 | 14  - 30  5 16  - 28 |  6
5 |  3 32  -  -  - 18  7 |  5
4 |  - 13  - 31  - 27  - |  4
3 | 23  2  -  -  -  8 19 |  3
2 | 12  - 24 21 10  - 26 |  2
1 |  1 22 11  - 25 20  9 |  1
+----------------------+
a  b  c  d  e  f  g

Board size = 7
Tour length = 49
Tie Breaker = procedure RothTieBreaker
a  b  c  d  e  f  g
+----------------------+
7 | 35 14 21 46  7 12  9 |  7
6 | 20 49 34 13 10 23  6 |  6
5 | 15 36 45 22 47  8 11 |  5
4 | 42 19 48 33 40  5 24 |  4
3 | 37 16 41 44 27 32 29 |  3
2 | 18 43  2 39 30 25  4 |  2
1 |  1 38 17 26  3 28 31 |  1
+----------------------+
a  b  c  d  e  f  g```

## J

Solution:
The Knight's tour essay on the Jwiki shows a couple of solutions including one using Warnsdorffs algorithm.

```NB. knight moves for each square of a (y,y) board